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This    book    sho'iitl   b<*   rriurncil    <»n    or   before    »hr   date   last    marked   belo 


PARTIAL 

DIFFERENTIAL  EQUATIONS 
OF  MATHEMATICAL  PHYSICS 

By  H.  BATEMAN 

This  truly  encyclopedic  exposition  of  the 
methods  of  solving  boundary  value  problems  of 
mathematical  physics  by  means  of  definite  ana 
lytical  expressions  is  valuable  as  both  text  and 
reference  work. 

It  covers  an  astonishingly  broad  field  of  prob- 
,  lems  and  contains  full  references  to  classical 
and  contemporary  literature,  as  well  as  numerous 
examples  on  which  the  reader  may  test  his  skill. 
This  edition  contains  a  number  of  corrections 
and  additional  references  furnished  by  the  late 
Professor  Bateman. 

The  book  includes  sections  on: 

Relation  of  the  differential  equations  to  varia 
tional  principles,  approximate  solution  of  bound- 
ary value  problems,  method  *of  Ritz,  orthogonal 
functions;  classical  equations,  including  uniform 
motion,  Fourier  series,  free  and  forced  vibrations, 
Heaviside's  expansion,  wave  motion,  potentials, 
Laplace's  equation. 

Applic?^ns  of  the  theorems  of  Green  and 
Stokes;  '  ><nann's  method,  elastic  solids,  fluid 
motion,  torsion,  membranes,  electromagnetism; 
two  dimensional  problems,  Fourier  inversion,  vi- 
bration of  a  loaded  string  and  of  a  shaft;  con- 
formal  mapping,  including  the  Riemann  theorem, 
the  distortion  theorem,  mapping  of  polygons. 

Equations  in  three  variables,  wave  motion, 
teat  flow;  polar  co-ordinates,  Legendre  polyno- 
nials,  with  applications;  cylindrical  co-ordinates, 
diffusion,  vibration  of  a  circular  membrane;  el- 
liptic and  parabolic  co-ordinates,  with  the  cor- 
responding boundary  problems;  torodial  co-ordi- 
nates and  applications, 

".  .  the  book  must  be  in  the  hands  of  every 
one  who  is  interested  in  the  boundary  value 
problems  of  mathematical  physics'*.  —  Bulletin 
of  American  Mathematical  Society. 

Text  in  English.  6x9.  xxii-f-522  pages,  29 
illustrations.  Originally  published  at  $10.50. 


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PARTIAL 
DIFFERENTIAL  EQUATIONS 

OF 

MATHEMATICAL  PHYSICS 


BY 

H.  BAT  EM  AN,  M.A.,  PH.D. 

Late  Fellow  of  Trinity  College,  Cambridge ; 

Professor  of  Mathematics,  Theoretical  Physics 

and  Aeronautics,  California  Institute  of  Technology, 

Pasadena,  California 


NEW  YORK 
DOVER  PUBLICATIONS 

1944 


First  Edition 1932 


First  American  Edition 1944 

By  special  arrangement  with  the 
Cambridge  University  Press  and  The  Macmillan  Co. 


Printed  in  the  U.  S.  A. 


Dedicated 

to 
MY  MOTHER 


CONTENTS 

PREFACE  page  xiii 

INTRODUCTION  xv-xxii 

CHAPTER  I 

THE  CLASSICAL  EQUATIONS 

§§  1-11-1-14.    Uniform  motion,  boundary  conditions,  problems,  a  passage  to  the 

limit.  1-7 

§§  1-15-1-19.  Fourier's  theorem,  Fourier  constants,  Cesaro's  method  of  summation, 
Parseval's  theorem,  Fourier  series,  the  expansion  of  the  integral  of  a  bounded 
function  which  is  continuous  bit  by  bit.  .  7-16 

§§  1-21-1-25.  The  bending  of  a  beam,  the  Green's  function,  the  equation  of  three 
moments,  stability  of  a  strut,  end  conditions,  examples.  16-25 

§§  1*31-1-36.  F^ee  undamped  vibrations,  simple  periodic  motion,  simultaneous 
linear  equations,  the  Lagrangian  equations  of  motion,  normal  vibrations,  com- 
pound pendulum,  quadratic  forms,  Hermit ian  forms,  examples.  25-40 

§§  1-41-1 -42.   Forced  oscillations,  residual  oscillation,  examples.  40-44 

§  1-43.  Motion  with  a  resistance  proportional  to  the  velocity,  reduction  to  alge- 
braic equations.  44  d7 

§  1-44.  The  equation  of  damped  vibrations,  instrumental  records.  47-52 

§  1-45-1 -46.   The  dissipation  function,  reciprocal  relations.  52-54 

§§  1-47-1-49.  Fundamental  equations  of  electric  circuit  theory,  Cauchy's  method 

of  solving  a  linear  equation,  Heaviside's  expansion.  54-6Q 

§§  1-51— 1-56.  The  simple  wave-equation,  wave  propagation,  associated  equations, 
transmission  of  vibrations,  vibration  of  a  building,  vibration  of  a  string,  torsional 
oscillations  of  a  rod,  plane  waves  of  sound,  waves  in  a  canal,  examples.  60-73 

§§  1-61-1 -63.  Conjugate  functions  and  systems  of  partial  differential  equations, 
the  telegraphic  equation,  partial  difference  equations,  simultaneous  equations 
involving  high  derivatives,  examplu.  73-77 

§§  1-71-1-72.  Potentials  and  stream-functions,  motion  of  a  fluid,  sources  and 
vortices,  two-dimensional  stresses,  geometrical  properties  of  equipotentials  and 
lines  of  force,  method  of  inversion,  examples.  77-90 

§§  1-81-1-82.  The  classical  partial  differential  equations  for  Euclidean  space, 
Laplace's  equation,  systems  of  partial  differential  equations  of  the  first  order 
fchich  lead  to  the  classical  equations,  elastic  equilibrium,  equations  leading  to  the 
^uations  of  wave-motion,  90-95 

S  1*91.   Primary  solutions,  Jacobi's  theorem,  examples.  95-100 

$1'92./  The  partial  differential  equation  of  the  characteristics,  bicharacteristics 

and  rays.  101-105 

;§§  1 '93-1  «94.  Primary  solutions  of  the  second  grade,  primitive  solutions  of  the 
wave-equation,  primitive  solutions  of  Laplace's  equation.  105-111 

§1-95.  Fundamental  solutions,  examples.  111-114 


viii  Contents 

CHAPTER  n 

APPLICATIONS  OF  THE  INTEGRAL  THEOREMS  OF  GREEN  AND  STOKES 

§§2*11-2-12.  Green's  theorem,  Stokes' s  theorem,  curl  of  a  vector,  velocity 
potentials,  equation  of  continuity.  pages  116-118 

§§  2-13-2-16.   The  equation  of  the  conduction  of  heat,  diffusion,  the  drying  of 

wood,  the  heating  of  a  porous  body  by  a  warm  fluid,  Laplace's  method,  example.     118-125 

§§  2-21-2*22.  Riemann's  method,  modified  equation  of  diffusion,  Green's  func- 
tions, examples.  126-131 

<§f  2-23-2*26.  Green' s^theorem^for  a  general  linear  differential  equation  of  the 
second  order,  characteristics,  classification  of  partial  differential  equations  of  the 
second  order,  a  property  of  equations  of  elliptic  type,  maxima  and  minima  of 
solutions.  131-138 

§§  2-31-2-32.  Green's  theorem  for  Laplace's  equation,  Green's  functions,  reciprocal 
relations.  ~  "  138-144 

§§  2-33-2-34.  Partial  difference  equations,  associated  quadratic  form,  the  limiting 

process,  inequalities,  properties  of  the  limit  function.  144-152 

§§  2-41-2-42.  The  derivation  of  physical  equations  from  a  variational  principle, 
Du  Bois-Reymond's  lemma,  a  fundamental  lemma,  the  general  Eulerian  rule, 
examples.  152-157 

§§  2-431-2-432.  The  transformation  of  physical  equations,  transformation  of 
Eulerian  equations,  transformation  of  Laplace's  equation,  some  special  trans- 
formations, examples.  157-162 

§  2-51.  The  equations  for  the  equilibrium  of  an  isotropic  elastic  solid.  162-164 

§  2-52.  The  equations  of  motion  of  an  inviscid  fluid.  164-166 

§  2-53.  The  equations  of  vortex  motion  and  Liouville's  equation.  .      166-169 

§  2-54.  The  equilibrium  of  a  soap  film,  examples.  169-171 

§§  2-56-2-56.  The  torsion  of  a  prism,  rectilinear  viscous  flow,  examples.  172-176 

§  2-57.  The  vibration  of  a  membrane.  176-177 

§§  2-58-2-59.  The  electromagnetic  equations,  the  conservation  of  energy  and 
momentum  in  an  electromagnetic  field,  examples.  177-183 

§§  2-61-2-62.  Kirchhoflfs  formula,  Poisson's  formula,  examples.  184-189 

§§  2-63-2-64.   Helmholtz's  formula,  Volterra's  method,  examples.  189-192 

§§  2-71-2-72.  Integral  equations  of  electromagnetism,  boundary  conditions,  the 
retarded  potentials  of  electromagnetic  theory,  moving  electric  pole,  moving  electric 
and  magnetic  dipoles,  example.  192-201 

§  2-73.  The  reciprocal  theorem  of  wireless  telegraphy.  201-203 


Contents 


IX 


CHAPTER  IH 
TWO-DIMENSIONAL  PROBLEMS 

§  3*11.  Simple  solutions  and  methods  of  generalisation  of  solutions,  example,  pages  204-207 
§3*12.  Fourier's  inversion  formula.  207-211 

§§3-13-3-15.   Method  of  summation,  cooling  of  fins,  use  of  simple  solutions  of  a 
complex  type,  transmission  of  vibrations  through  a  viscous  fluid,  fluctuating  tem- 
peratures and  their  transmission  through  the  atmosphere,  examples.  211-215 
§  3-16.  Poisson's  identity,  examples.  215-218 
§  3*17.  Conduction  of  heat  in  a  moving  medium,  examples.  218—221 
§  3-18.   Theory  of  the  unloaded  cable,  roots  of  a  transcendental  equation,  Kosh- 
liakov's  theorem,  effect  of  viscosity  on  sound  waves  in  a  narrow  tube.                       221-228 
§  3-21.    Vibration  of  a  light  string  loaded  at  equal  intervals,  group  velocity, 
electrical  filter,  torsional  vibrations  of  a  shaft,  examples.  228-236 
§§  3-31-3-32.    Potential  function  with  assigned  values  on  a  circle,  elementary 
treatment  of  Poisson's  integral,  examples.  236-242 
§§  3-33-3-34.  Fourier  series  which  are  conjugate,  Fatou's  theorem,  Abel's  theorem 
for  power  series.  242-245 
§  3-41.  The  analytical  character  of  a  regular  logarithmic  potential.                  •         245-246 
§  3-42.  Harnack's  theorem.                                                                                                246-247 
§  3-51.  Schwarz's  alternating  process.                                                                            247-249 
§  3-61.  Flow  round  a  circular  cylinder,  examples.                                                          249-254 
§  3-71.    Elliptic  coordinates,  induced  charge  density,  Munk's   theory  of  thin 
aerofoils.                                                                                                                               254r~260 
§§  3-81-3-83.    Bipolar  co-ordinates,  effect  of  a  mound  or  ditch  on  the  electric 
potential,  example,  the  effect  of  a  vertical  wall  on  the  electric  potential.                   260-265 


CHAPTER  IV 
CONFORMAL  REPRESENTATION 

§§  4-1 1-4-21.   Properties  of  the  mapping  function,  invariants,  Riemann  surfaces 
and  winding  points,  examples.  266-270 

§§  4-22—4-24.  The  bilinear  transformation,  Poisson's  formula  and  the  mean  value 
theorem,  the  conformal  representation  of  a  circle  on  a  half  plane,  examples.  270-275 

§§  4-31-4-33.   Riemann' s  problem,  properties  of  regions,  types  of  curves,  special 
and  exceptional  cases  of  the  problem.  275-280 

§§  4-41-4-42.  The  mapping  of  a  unit  circle  on  itself,  normalisation  of  the  mapping 
problem,  examples.  280-283 

§  4-43.  The  derivative  of  a  normalised  mapping  function,  the  distortion  theorem 
and  other  inequalities.  283-285 

§  4-44.  The  mapping  of  a  doubly  carpeted  circle  with  one  interior  branch  point.    285-287 
§4-45.  The  selection  theorem.  287-291 

§4-46.  Mapping  of  an  open  region.  291-292 

b-2 


x  Contents 

§  4-51.  The  Green's  function.  pages  292-294 

§§  4-61-4-63.  Schwarz's  lemma,  the  mapping  function  for  a  polygon,  mapping  of 

a  triangle,  correction  for  a  condenser,  mapping  of  a  rectangle,  example.  294-305 

§  4-64.  Conformal  mapping  of  the  region  outside  a  polygon,  example.  305-309 

§§  4-71-4-73.    Applications  of  conformal  representation  in  hydrodynamics,  the 
mapping  of  a  wing  profile  on  a  nearly  circular  curve,  aerofoil  of  small  thickness.     309-316 

§§  4-81-4-82.    Orthogonal  polynomials  connected  with  a  given  closed  curve,  the 
mapping  of  the  region  outside  C',  examples.  316-322 

§§  4-91-4-93.   Approximation  t*>  the  mapping  function  by  means  of  polynomials, 
Daniell's  orthogonal  polynomials,  Fe JOT'S  theorem.  322-328 

CHAPTER  V 
EQUATIONS  IN  THREE  VARIABLES 

§§  5-11-5-12.     Simple    solutions  and    their    generalisation,   progressive  waves, 
standing  waves,  example.  329-331 

§  5-13.   Reflection  and  refraction  of  electromagnetic  waves,  reflection  and  refrac- 
tion of  plane  waves  of  sound,  absorption  of  sound,  examples.  331-338 

§  5-21.   Some  problems  in  the  conduction  of  heat.  338-345 

§  5-31.   Two-dimensional  motion  of  a  viscous  fluid,  examples.  345-350 


CHAPTER  VI 

POLAR  CO-ORDINATES 

§§  6-11-6-13.   The  elementary  solutions,  cooling  of  a  solid  sphere.  351-354 

§§  6-21-6-29.  Legendre  functions,  Hobson's  theorem,  potential  function  of  degree 
zero,  Hobson's  formulae  for  Legendre  functions,  upper  and  lower  bounds  for  the 
function  Pn  (/-i),  expressions  for  the  Legendre  functions  as  nth  derivatives,  the 
associated  Legendre  functions,  extensions  of  the  formulae  of  Rodrigues  and 
Conway,  integral'  relations,  properties  of  the  Legendre  coefficients,  examples.  354-366 

§§  6-31—6-36.  Potential  function  with  assigned  values  on  a  spherical  surface  $, 
derivation  of  Poisson's  formula  from  Gauss's  mean  value  theorem,  some  applica- 
tions of  Gauss's  mean  value  theorem,  the  expansion  of  a  potential  function  in  a 
series  of  spherical  harmonics,  Legendre's  expansion,  expansion  of  a  polynomial  in 
a  series  of  surface  harmonics.  367-375 

§§  6-41-6-44.  Legendre  functions  and  associated  functions,  definitions  of  Hobson 
and  Barnes,  expressions  in  terms  of  the  hypergeometric  function,  relations  be- 
tween the  different  functions,  reciprocal  relations,  potential  functions  of  degree 
n  +  \  where  n  is  an  integer,  conical  harmonics,  Mehler's  functions,  examples.  375-384 

§§  6-51-6-54.  Solutions  of  the  wave-equation,  Laplace's  equation  in  n  +  2  variables, 
extension  of  the  idea  of  solid  angle,  diverging  waves,  Hankel's  cylindrical 
functions,  the  method  of  Stieltjes,  Jacobi's  polynomial  expansions,  Wangerin's 
formulae,  examples.  384-395 

§  6-61.  Definite  integrals  for  the  Legendre  functions.  395-397 


Contents  xi 


CHAPTER  VII 
CYLINDRICAL  CO-ORDINATES 

§  7*11.  The  diffusion  equation  in  two  dimensions,  diffusion  from  a  cylindrical  rod, 
examples.  pages  398-399 

§  7-12.  Motion  of  an  incompressible  viscous  fluid  in  an  infinite  right  circular 
cylinder  rotating  about  its  axis,  vibrations  of  a  disc  surrounded  by  viscous  fluid, 
examples.  399-401 

§  7«  13.  Vibration  of  a  circular  membrane.  401 

§7-21.  The  simple  solutions  of  the  wave-equation,  properties  of  the  Bessel 
functions,  examples.  402-404 

§  7-22.  Potential  of  a  linear  distribution  of  sources,  examples.  404-405 

§  7-31.  Laplace's  expression  for  a  potential  function  which  is  symmetrical  about  an 
axis  and  finite  on  the  axis,  special  cases  of  Laplace's  formula,  extension  of  the 
formula,  examples.  405-409 

§  7-32.  The  use  of  definite  integrals  involving  Bessei  functions,  Sommerf eld's 
expression  for  a  fundamental  wave-function,  Hankel's  inversion  formula, 
examples.  409-412 

§  7-33.  Neumann's  formula,  Green's  function  for  the  space  between  two  parallel 

planes,  examples.  *  412-415 

§  7-41.  Potential  of  a  thin  circular  ring,  examples.  416-417 

§  7*42.  The  mean  value  of  a  potential  function  round  a  circle.  418-419 

§  7-51.  An  equation  which  changes  from  the  elliptic  to  the  hyperbolic  type.  419-420 

CHAPTER  VIII 
ELLIPSOIDAL  CO-ORDINATES 

§§  8-1 1-8-12.  Confocal  co-ordinates,  special  potentials,  potential  of  a  homogeneous 

solid  ellipsoid,  Maclaurin's  theorem.  421-425 

§  8-21 .  Potential  of  a  solid  hypersphere  whose  density  is  a  function  of  the  distance 

from  the  6entre.  426-427 

§§  8-31-8-34.  Potential  of  a  homoeoid  and  of  an  ellipsoidal  conductor,  potential 
of  a  homogeneous  elliptic  cylinder,  elliptic  co-ordinates,  Mathieu  functions, 
examples.  427-433 

§§  8-41-8-45.  Prolate  spheroid,  thin  rod,  oblate  spheroid,  circular  disc,  con- 
ducting ellipsoidal  column  projecting  above  a  flat  conducting  plane,  point  charge 
above  a  hemispherical  boss,  point  charge  in  front  of  a  plane  conductor  with  a  pit 
or  projection  facing  the  charge.  433-439 

§§  8-51-8-54.  Laplace's  equation  in  spheroidal  co-ordinates,  Lame  products  for 
spheroidal  co-ordinates,  expressions  for  the  associated  Legendre  functions, 
spheroidal  wave-functions,  use  of  continued  fractions  and  of  integral  equations, 
a  relation  between  spheroidal  harmonics  of  different  types,  potential  of  a  disc, 
examples.  440-448 


Xll 


Contents 


CHAPTER  IX 

PARABOLOIDAL  CO-ORDINATES 

§  9-11.  Transformation  of  the  wave-equation,  Lam6  products.  pages  449-451 

§§  0-21-0-22.   Sonine's  polynomials,  recurrence  relations,  roots,  orthogonal  pro- 
perties, Hermite's  polynomial,  examples.  451-455 
§  0-31.    An  erpression  for  the  product  of  two  Sonine  polynomials,  confluent 
hypergeometric  functions,  definite  integrals,  examples.                                             455-460 

CHAPTER  X 

TOROIDAL  COORDINATES 

§  10-1 .  Laplace's  equation  in  toroidal  co-ordinates,  elementary  solutions,  examples.  461-462 
§  10-2.  Jacobi's  transformation,  expressions  for  the  Legendre  functions,  examples.  463-465 
§  10-3.  Green's  functions  for  the  circular  disc  and  spherical  bowl.  465-468 
§  10-4.  Relation  between  toroidal  and  spheroidal  co-ordinates.  468 
§  10-5.  Spherical  lens,  use  of  the  method  of  images,  stream-function.  468-472 
§§  10-6-10-7.  The  Green's  function  for  a  wedge,  the  Green's  function  for  a  semi- 
infinite  plane.  472-474 
§  10-8.  Circular  disc  in  any  field  of  force.  474-475 


CHAPTER  XI 
DIFFRACTION  PROBLEMS 

§§  11-1-11-3.   Diffraction  by  a  half  plane,  solutions  of  the  wave-equation,  Som- 

merfeld's  integrals,  waves  from  a  line  source,  Macdonald's  solution,  waves  from 

a  moving  source. 

§  11-4.  Discussion  of  Sommerf eld's  solution. 

§§  11 -5-1 1-7.  Use  of  parabolic  co-ordinates,  elliptic  co-ordinates. 


476-483 
483-486 
486-490 


CHAPTER  XII 
NON-LINEAR  EQUATIONS 

§12*1.    Riccati's  equation,  motion  of  a  resisting  medium,  fall  of  an  aeroplane, 
bimolecular  chemical  reactions,  lines  of  force  of  a  moving  electric  pole,  examples.     491-496 
§  12-2.   Treatment  of  non-linear  equations  by  a  method  of  successive  approxi- 
mations, combination  tones,  solid  friction.  497-501 
§  12*3.    The  equation  for  a  minimal  surface,   Plateau's  problem,  Schwarz's 
method,  helicoid  and  catenoid.  501-509 
§  12-4.  The  steady  two-dimensional  motion  of  a  compressible  fluid,  examples.         509-511 


APPENDIX 

LIST  OF  AUTHORS  CITED 

INDEX 


512-514 
515-519 
520-522 


PREFACE  TO  AMERICAN  EDITION 

The  publishers  are  gratified  that  through  the  cooperation  of  The 
Macmillan  Company  and  the  Cambridge  University  Press,  Professor 
Bateman's  definitive  work  on  partial  differential  equations  is  made 
available  for  text  and  reference  use  by  American  mathematicians  and 
physicists  in  a  reduced  price,  corrected  edition. 

Professor  Batemanthas  kindly  provided  the  corrections  and  addi- 
tions to  references  which  are  found  on  pages  11,  15,  24,  49,  50,  73, 
124,  126,  127,  212,  219,  229,  230,  247,  261,  359,  364,  366,  403,  404, 
410,  415,  417,  452,  454,  457,  462,  464,  477,  515,  517,  518  and  519. 
The  correction  required  on  page  219  was  brought  to  Professor 
Bateman's  attention  by  Mr.  W.  H.  Jurney. 

DOVER  PUBLICATIONS 


PREFACE 

IN  this  book  the  analysis  has  been  developed  chiefly  with  the  aim  of 
obtaining  exact  analytical  expressions  for  the  solution  of  the  boundary 
problems  of  mathematical  physics. 

In  many  cases,  however,  this  is  impracticable,  and  in  recent  years  much 
attention  has  been  devoted  to  methods  of  approximation.  Since  these  are 
not  described  in  the  text  with  the  fullness  which  they  now  deserve,  a  brief 
introduction  has  been  written  in  which  some  of  these  methods  are  sketched 
and  indications  are  given  of  portions  of  the  text  which  will  be  particularly 
useful  to  a  student  who  is  preparing  to  use  these  methods. 

No  discussion  has  been  given  of  the  partial  differential  equations  which 
occur  in  the  new  quantum  theory  of  radiation  because  these  have  been  well 
treated  in  several  recent  book^,  and  an  adequate  discussion  in  a  book  of 
this  type  would  have  greatly  increased  its  size.  It  is  thought,  however,  that 
some  of  the  analysis  may  prove  useful  to  students  of  the  new  quantum 
theory. 

Some  abbreviations  jmd  slight  departures  from  the  notation  used  in 
recent  books  have  been  adopted.  Since  the  /,-notation  for  the  generalised 
Laguerre  polynomial  has  been  used  recently  by  different  writers  with 
slightly  different  meanings,  the  original  T-notation  of  Sonine  has  been 
retained  as  in  the  author's  Electrical  find  Optical  Wave  Motion.  It  is 
thought,  however,  that  a  standardised  //-notation  will  eventually  be 
adopted  by  most  writers  in  honour  of  the  work  of  Lagrange  and  Laguerre. 

The  abbreviations  '"eit"  and  "eif  "  used  in  the  text  might  be  used  with 
advantage  in  the  new  quantum  theory,  together  with  some  other  abbrevia- 
tions, such  as  "oil"  for  eigenlrsrei  and  k'eiv"  for  eigenvector. 

The  Heaviside  Calculus  and  Ihe  theory  of  integral  equations  are  only 
briefly  mentioned  in  the  text:  they  belong  rather  to  a  separate  subject 
which  might  be  called  the  Integral  Equations  of  Mathematical  Physics. 
Accounts  of  the  existence  theorems  of  potential  theory,  Sturm-Liouville 
expansions  and  ellipsoidal  harmonics  have  also  been  omitted.  Many 
excellent  books  have  however  appeared  recently  in  which  these  subjects 
are  adequately  treated. 

I  feel  indeed  grateful  to  the  Cambridge  University  Press  for  their  very 
accurate  work  and  intelligent  assistance  during  the  printing  of  this  book. 

M.  BATEMAN 
November  1931 


INTRODUCTION 

THE  differential  equations  of  mathematical  physics  are  now  so  numerous 
and  varied  in  character  that  it  is  advisable  to  make  a  choice  of  equations 
when  attempting  a  discussion. 

The  equations  considered  in  this  book  are,  I  believe,  all  included  in  some 
set  of  the  form          x  «  x  ~  <>  F 

—  -  0  -  0  -  -  0 

to     '  ¥.'   "•    to.     ' 

where  the  quantities  on  the  left-hand  sides  of  these  equations  are  the 
variational  derivatives*  of  a  quantity  F9  which  is  a  function  of  I  independent 
variables  xlt  ...  xlt  of  m  dependent  variables  yl9  "...  ym  and  of  the  deriva- 
tives up  to  order  n  of  the  t/'s  with  respect  to  the  x's.  The  meaning  of  a 
variational  derivative  will  be  gradually  explained. 

(T)  The  first  property  to  be  noted  is  that  the  variational  derivatives  of  a 
function  F  all  vanish  identically  when  the  function  can  be  expressed  in  the 

form  F_dol    dGt         dat 

JL  —  j       r  ~j        r  •  •  •    i    j      j 
dx-i      cLx^  axi 

where  each  of  the  functions  G8  is  a  function  of  the  x's  and  t/'s  and  of  the 
derivatives  up  to  order  n  —  1  of  the  i/'s  with  respect  to  the  x'a.  The  notation 
d/dx8  is  used  here  for  a  complete  differentiation  with  respect  to  x8  when 
consideration  is  taken  of  the  fact  that  F  is  not  only  an  explicit  function  of 
xs  but  also  an  explicit  function  of  quantities  which  are  themselves  functions 
of  x8. 

Another  statement  of  the  property  just  mentioned  is  that  the  varia- 
tional derivatives  of  F  vanish  identically  when  the  expression 

F  dxldx2  ...  dxl 
is  an  exact  differential. 

In  the  case  when  there  is  only  one  independent  variable  x  and  only  one 
dependent  variable  y,  whose  derivatives  up  to  order  n  are  respectively 
y'>  y"y  •••  y(n)9  the  condition  that  Fdx  may  be  an  exact  differential  is 
readily  found  to  be 


- 
dy      dx  \dy'        dx*\dy" 

Now  the  quantity  on  the  left-hand  side  of  this  equation  is  indeed  the 
variational  derivative  of  F  with  respect  to  y  and  will  be  denoted  by  the 
symbol  SF/Sy. 

*  For  a  systematic  discussion  of  variational  derivatives  reference  may  be  made  to  the  papers 
of  Th.  de  Donder  in  Bulletin  de  VAccuttmie  BoyaU  de  Bdgique,  Claase  des  Sciences  (5),  t.  xv 
(1929-30).  In  some  cases  a  set  of  equations  must  be  supplemented  by  another  to  give  all  the 
equations  in  a  set  of  the  variational  form. 


xvi  Introduction 

In  the  case  when  F  is  of  the  form 

yN.  (z)  -  zMx  (y), 

where  z  is  a  function  of  x  and  Mx  (y),  Nx  (z)  are  linear  differential  expres- 
sions involving  derivatives  up  to  order  n  and  coefficients  of  these  deriva- 
tives which  are  functions  of  x  with  a  suitable  number  of  continuous 
derivatives,  we  can  say  that  the  differential  expressions  Mx  (y),  Nx  (z)  are 
adjoint  when  8F/8y  ~  0  for  all  forms  of  the  function  z.  When  z  is  chosen 
to  be  a  solution  of  the  differential  equation  Nx  (z)  =  0  the  expression 
zM  x  (y)  dx  is  an  exact  differential  and  so  z  is  an  integrating  factor  of  the 
differential  equation  adjoint  to  Nx  (z)  =  0.  The  relation  between  two 
adjoint  differential  equations  is,  moreover,  a  reciprocal  one, 

The  idea  of  adjoint  differential  expressions  was  introduced  by  Lagrange 
and  extended  by  Riemann  to  the  case  when  there  is  more  than  one  inde- 
pendent variable.  Further  extensions  have  been  made  by  various  writers 
for  the  case  when  there  are  several  dependent  variables*.  Adjoint 
differential  expressions  and  adjoint  differential  equations  are  now  of  great 
importance  in  mathematical  analysis. 

A  second  important  property  of  the  variational  derivatives  may  be 
introduced  by  first  considering  a  simple  integral 


and  its  first  variation 


Integrating  the  (s-f  l)th  term  s  times  by  parts,  making  use  of  the 
equations 


it  is  readily  seen  that  the  portion  of  87  which  still  remains  under  the  sign  of 
integration  is 


It  is  readily  understood  now  why  the  name  "  variational  derivative  "  is 
used.  The  variational  derivative  is  of  fundamental  importance  in  the 
Calculus  of  Variations  because  the  Eulerian  differential  equation  for  a 
variational  problem  involving  an  integral  of  the  above  form  is  obtained  by 
equating  the  variational  derivative  to  zero. 

This  rule  is  capable  of  extension,  and  rules  for  writing  down  the 
variational  derivatives  of  a  function  F  in  the  general  case  when  there  are 

*  See  for  instance  J.  Kiirechak,  Math.  Ann.  Bd.  Lxn,  S.  148  (1906);  D.  R.  Davis,  Trans. 
Amer.  Math.  Soc.  vol.  xxx,  p.  710  (1928). 


Introduction  xvii 

I  independent  variables  and  m  dependent  variables  can  be  derived  at  once 
from  the  rules  of  §  2-42  for  the  derivation  of  the  Eulerian  equations. 

Since  our  differential  equations  are  always  associated  with  variational 
problems,  direct  methods  of  solving  these  problems  are  of  great  interest. 

The  important  method  of  approximation  invented  by  Lord  Rayleigh* 
and  developed  by  W.  Tlitzf  is  only  briefly  mentioned  in  the  text,  though  it 
has  been  used  by  RitzJ,  Timoshenko§  and  many  other  writers|!  to  obtain 
approximate  solutions  of  many  important  problems.  An  adequate  dis- 
cussion by  means  of  convergence  theorems  is  rather  long  and  difficult,  and 
has  been  omitted  from  the  text  largely  for  this  reason  and  partly  because 
important  modifications  of  the  method  have  recently  been  suggested  which 
lead  more  rapidly  to  the  goal  and  furnish  means  of  estimating  the  error  of 
an  approximation. 

In  Ritz's  method  a  boundary  problem  for  a  differential  equation 
D  (u)  =  0  is  replaced  by  a  variation  problem  in  which  a  certain  integral  / 
is  to  be  made  a  minimum,  the  unknown  function  u  being  subject  to  certain 
supplementary  conditions  which  are  usually  linear  boundary  conditions 
and  conditions  of  continuity.  The  function  ua,  used  by  Ritz  as  an  approxi- 
mation for  u,  is  not  generally  a  solution  of  the  differential  equation,  but  it 
does  satisfy  the  boundary  conditions  for  all  values  of  the  arbitrary  con- 
stants which  it  contains.  The  result  is  that  when  an  integral  Ia  is  calculated 
from  ua  in  the  way  that  /  is  to  be  calculated  from  u,  the  integral  Ia  is 
greater  than  the  minimum  value  Im  of  /,  even  when  the  arbitrary  con- 
stants in  ua  are  chosen  so  as  to  make  Ia  as  small  as  possible.  This  means 
that  Im  is  approached  from  above  by  integrals  of  the  type  Ia . 

Now  it  was  pointed  out  by  R.  Courant  ^[  that  Im  can  often  be  approached 
from  below  by  integrals  Ib  calculated  from  approximation  functions  ub 
which  satisfy  the  differential  equation  but  are  subject  to  less  restrictive 
supplementary  conditions.  If,  for  instance,  u  is  required  to  be  zero  on  the 
boundary,  the  boundary  condition  may  be  loosened  by  merely  requiring 
ub  to  give  a  zero  integral  over  the  boundary  in  each  of  the  cases  in  which  it 
is  first  multiplied  by  a  function  vs  belonging  to  a  certain  finite  set.  This  idea 
has  been  developed  by  Trefftz**  who  uses  the  arithmetical  mean  of  Ia  and  Ib 

*  Phil.  Trans.  A,  vol.  CLXI,  p.  77  (1870);  Scientific  Papers,  vol.  I,  p.  57. 

t  W.  Ritz,  Crette,  Bd.  cxxxv,  S.  1  (1908);  (Euvres,  pp.  192-316  (Gauthier-Villars,  Paris,  1911). 

J  W.  Ritz,  Ann.  der  Phys.  (4),  Bd.  xxvm,  S.  737  (1909). 

§  8.  Timoshcnko,  Phil.  Mag.  (6),  vol.  XLVII,  p.  1093  (1924);  Proc.  London  Math.  Soc.  (2), 
vol.  XX,  p.  398  (1921);  Trans.  Amcr.  Soc.  Civil  Engineers,  vol.  LXXXVII,  p.  1247  (1924);  Vibration 
Problems  in  Engineering  (D.  Van  Nostrand,  New  York,  1928). 

||  See  especially  M.  Plancherel,  Bull,  dcs  Sciences  Math.  t.  XLVII,  pp.  376,  397  (1923),  t.  XLVIII, 
pp.  12,  58,  93  (1924);  Comptcs  JRcndus,  t.  CLXIX,  p.  1152  (1919);  R.  Courant,  Acia  Math.  t.  XLIX, 
p.  1  (1926);  K.  Friedrichs,  Math.  Ann.  Bd.  xcvni,  S.  205  (1927-8). 

U  R.  Courant,  Math.  Ann.  Bd.  xcvn,  S.  711  (1927). 

**  E.  Trefftz,  Int.  Congress  o£  Applied  Mechanics,  Zurich  (1926),  p.  135;  Math.  Ann.  Bd.  c, 
S.  603  (1928). 


xviii  Introduction 

as  a  close  approximation  for  Im  ,  and  uses  the  difference  of  Ia  and  Ib  as  an 
upper  bound  for  the  error  in  this  method  of  approximation.  This  method  is 
simplified  by  a  choice  of  functions  v8  which  will  make  it  possible  to  find 
simple  solutions  of  the  differential  equation  for  the  loosened  boundary 
conditions.  Sometimes  it  is  not  the  boundary  conditions  but  the  conditions 
of  continuity  which  should  be  loosened,  and  this  makes  it  advisable  not  to 
lose  interest  in  a  simple  solution  of  a  differential  equation  because  it  does 
not  satisfy  the  requirements  of  continuity  suggested  by  physical  con- 
ditions. 

In  order  that  Ritz's  method  may  be  used  we  must  have  a  sequence  of 
functions  which  satisfy  the  boundary  conditions  and  conditions  of  con- 
tinuity peculiar  to  the  problem  in  hand.  It  is  advantageous  also  if  these 
functions  can  be  chosen  so  that  they  form  an  orthogonal  set,  To  explain 
what  is  meant  by  this  we  consider  for  simplicity  the  case  of  a  single 
independent  variable  x.  The  functions  0!  (#),  ^2  (#)>  •••>  defined  in  an 
interval  a  <  x  <  6,  are  then  said  to  form  a  normalised  orthogonal  set  when 
the  orthogonal  relations 


=  1,     ra=  n 

are  satisfied  for  each  pair  of  functions  of  the  set.  This  definition  is  readily 
extended  to  the  case  of  several  independent  variables  and  functions 
defined  in  a  domain  R  of  these  variables;  the  only  difference  is  that  the 
simple  integrals  are  replaced  by  integrals  over  the  domain  of  definition.  The 
definition  may  be  extended  also  to  complex  functions  i/rn  (x)  of  the  form 
an  (x)  -f  ifin  (#)>  where  an  (x)  and  /?n  (x)  are  real.  The  orthogonal  relations 
are  then  of  type 


s  f 

Ja 


(x)  +  ipm  (x)]  [an  (x)  -  ipn  (x)]  dx=0,    m*n, 

=  1,    m  =  n. 

Many  types  of  orthogonal  functions  are  studied  in  this  book.  The 
trigonometrical  functions  sin  (nx)9  cos  (nx)  with  suitable  factors  form  an 
orthogonal  set  for  the  interval  (0,  2?r),  the  Legendre  functions  Pn  (x),  with 
suitable  normalising  factors,  form  an  orthogonal  set  for  the  interval 
(  —  1,  1),  while  in  Chapter  ix  sets  of  functions  are  obtained  .which  are 
orthogonal  in  an  infinite  interval.  The  functions  of  Laplace,  which  form  the 
complete  system  of  spherical  harmonics  considered  in  Chapter  vi,  give  an 
orthogonal  set  of  functions  for  the  surface  of  a  sphere  of  unit  radius,  and  it 
is  easy  to  construct  functions  which  are  orthogonal  in  the  whole  of  space. 
In  Chapter  iv  methods  are  explained  by  which  sets  of  normalised  orthogonal 
functions  may  be  associated  with  a  given  curve  or  with  a  given  area.  In 
many  cases  functions  suitable  for  use  in  Ritz's  method  of  approximation 


Introduction  xix 

are  furnished  by  the  Lam6  products  defined  in  Chapters  m-xi.  These  pro- 
ducts are  important,  then,  for  both  the  exact  and  the  approximate  solution 
of  problems.  It  was  shown  by  Ritz,  moreover,  that  sometimes  the  functions 
occurring  in  the  exact  solution  of  one  problem  may  be  used  in  the  ap- 
proximate solution  of  another;  the  functions  giving  the  deflection  of  a 
clamped  bar  were  in  fact  used  in  the  form  of  products  to  represent  the 
approximate  deflection  of  a  clamped  rectangular  plate. 

Early  writers*  using  Ritz's  method  were  content  to  indicate  the  degree 
of  approximation  obtainable  by  applying  the  method  to  problems  which 
could  be  solved  exactly  and  comparing  the  approximate  solution  with  the 
exact  solution.  This  plan  is  somewhat  unsatisfactory  because  the  examples 
chosen  may  happen  to  be  particularly  favourable  ones.  Attempts  have, 
however,  been  made  by  Krylofff  and  others  to  estimate  the  error  when  an 
approximating  function  of  order  n,  say 

Ua  =  $0  (%)  +  Cl<Al  (X)  +  C2^2  (X)+   ..*  +  Cntn  0*0, 

is  substituted  in  the  integral  to  be  minimised  and  the  coefficients  cs  are 
chosen  so  as  to  make  the  resulting  algebraic  expression  a  minimum. 
Attempts  have  been  made  also  to  determine  the  order  n  needed  to  make 
the  error  less  than  a  prescribed  quantity  e. 

,-  In  Ritz's  method  a  boundary  problem  for  a  given  differential  equation 
must  first  of  all  be  replaced  by  a  variation  problerp.  There  are,  however, 
modifications  of  Ritz's  method  in  which  this  step  is  avoided.  If,  for  in- 
stance, the  differential  equation  is  a  variational  equation  8F/8u  =  0,  the 
same  set  of  equations  for  the  determination  of  the  constants  cs  is  obtained 
by  substituting  the  expression  Ua  for  u  directly  in  the  equation 

tb 

Su  .  (SF/Su)  dx  =  0, 

J a 

and  equating  to  zero  the  coefficients  of  the  variations  8cs . 

This  method  has  been  recommended  by  HenckyJ  and  Goldsbrough§ ;  it 
has  the  advantage  of  indicating  a  reason  why  in  the  limit  the  function  ua 
should  satisfy  the  differential  equation. 

Another  method,  proposed  by  Boussinesq||  many  years  ago,  has  been 
called  the  method  of  least  squares.  If  the  differential  equation  is 

L9(u)  =  f(x), 
and  a  <  x  <  b  is  the  range  in  which  it  is  to  be  satisfied  with  boundary 

*  See,  for  instance,  M.  Paschoud,  Sur  V application  de  la  mtihode  de  W.  Ritz:  These  (Gauthier- 
Villars,  Paris,  1914). 

f  N.  Kryloff,  Comptes  Rendus,  t.  CLXXX,  p.  1316  (1925),  t.  CLXXXVI,  p.  298  (1928);  Annales 
de  Toulouse  (3),  t.  xix,  p.  167  (1927). 

J  H.  Hencky,  Zeits.fur  angew.  Math.  u.  Mech.  Bd.  TO,  S.  80  (1927). 

§  G.  R.  Goldsbrough,  Phil.  Mag.  (7),  vol.  vn,  p.  333  (1929). 

||  J.  Boussinesq,  Ttieorie  de  la  chaleur,  1. 1,  p.  316. 


xx  Introduction 

conditions  at  the  ends,  the  constants  c8  in  an  approximating  function  ua, 
which  satisfies  these  boundary  conditions,  are  chosen  so  as  to  make  the 
integral 

[Lm(ua)-f(x)]*dx 

J  a 

as  small  as  possible.  The  accuracy  of  this  method  has  been  studied  by 
Kryloff*  who  believes  that  Ritz's  method  and  the  method  of  least  squares 
are  quite  comparable  in  usefulness.  The  method  of  least  squares  is,  of 
course,  closely  allied  to  the  well-known  method  of  approximating  to  a 
function  /  (x)  by  a  finite  series  of  orthogonal  functions 


the  coefficients  cs  being  chosen  so  that  the  integral")* 

rb 

If  (*)  -  Ci<Ai  0*0  -  C2</r2  (X)  -  ...  ~  Cnifjn  (X)]2  dx 


may  be  as  small  as  possible.  The  conditions  for  a  minimum  lead  to  the 
equations  & 

'.=      f(s)j.(x)d8     («=l,2,...n). 

Ja 

For  an  account  of  such  methods  of  approximation  reference  may  be 
made  to  recent  books  by  Dunham  Jackson  J,  S.  Bernstein§  and  dc  la  Vallee 
Poussin||. 

In  the  discussion  of  the  convergence  of  methods  of  approximation 
there  is  an  inequality  due  to  Bouniakovsky  and  Schwarz  which  is  of 
fundamental  importance.  If  the  functions  f  (x),  g  (x)  and  the  parameter  c 
are  all  real,  the  integral 

f  [/  (x)  +  ^  (#)]2  =  A  -f  2cH  +  c2B 

Ja 

is  never  negative  and  so  AB  —  H2  >  0.  This  gives  the  inequality 
f  [/  (x)Y  dx  'b  \jy  (x)Y  dx  >  f  [/  (x)  g  (x)Y  dx. 

J  a  *  a  J  a 

There  is  a  similar  inequality  for  two  complex  functions  f(x),g  (x)  and 

*  N.  Kryloff,  Comptes  Rend  as,  t.  CLXI,  p.  558  (1915);  t.  CLXXXI,  p.  86  (11)25).  Sec  also  Krawt- 
chouk,  ibid.  t.  c'LXXxm,  pp.  474,  992  (1926). 

t  G.  I'larr,  Comptes  Rcttilus,  t.  XLIV,  p.  984  (1857);  A*  Tocpler,  Arizctycr  dcr  Kais.  Akad.  zu 
Wu'n  (1870),  p.  205. 

|  1).  Jackson,  "The  Theory  of  Approximation,"  Arner.  Math.  tioc.  Colloquium  publications, 
vol.  xi  (1930). 

§  S.  Hernstt'in,  Lv^ona  sur  le#  proprn'te*  extremal?*  rt  In  mcilleure  approximation  dcs  fotictions 
unalytiqiu's  <T  une  van  able  reelle  (Gauthier-Villars,  Paris,  1926). 

||  C.  J  .  do  la  Vallee  Poussm,  Lemons  sur  I'  approximation  des  foncttons  d'une  variable  reelle  (ibid. 
1919). 


Introduction  xxi 

their  conjugates  /  (x),  g  (x).    Indeed,  if  c  and  c  are  conjugate  complex 
quantities,  the  integral 


is  never  negative.  Writing 

f=l+im,     g=p+iq,     c=£+iri, 
where  I,  m,  p,  q,  £,  77  are  all  real,  the  integral  may  be  written  in  the  form 

A  (£2  +  r?2)  +  2B£-f  2^  +  D=  ~[(A£  +  £)*+  (A*i  +  < 
where  A  =       (p2  +  q2)  dx  =      gg  dx, 

J  a  Ja 

D  =  f  (i2  +  m2)  dx  =  (  //da;, 

'  Ja  J  a 

B  +  iC  =  f  fgdx,     B  -  iC  =  [ 

Ja  Ja 

Since  the  integral  is  never  negative,  we  have  the  inequality 

AD  >  B*  +  C2, 
which  may  be  written  in  the  form* 


In  this  inequality  the  functions  /  and  g  may  be  regarded  as  arbitrary 
integrable  functions.  This  inequality  and  the  analogous  inequality  for 
finite  sums  are  used  in  §  4-81. 

In  the  approximate  treatment  of  problems  in  vibration  the  natural 
frequencies  are  often  computed  with  the  aid  of  isoperimetric  variation 
problems.  Ritz's  method  is  now  particularly  useful.  If,  for  instance,  the 
differential  equation  is 


and  the  end  conditions      u  (a)  =  0,     u  (b)  =  0, 
the  aim  is  to  make  the  integral 


a  minimum  when  the  integral 

f  [«(*)]•  «fe 

Ja 

*  This  inequality  is  called  Schwarz's  inequality  by  E.  Schmidt,  Rend.  Palermo,  t.  xxv,  p.  58 
(1908). 


xxii  Introduction 

has  an  assigned  value.  This  is  accomplished  by  replacing  u  by  a  finite  series 
in  both  integrals  and  reducing  the  problem  to  an  algebraic  problem.  It  was 
noted  by  Rayleigh  that  very  often  a  single  term  in  the  series  will  give  a 
good  approximation  to  the  frequency  of  the  fundamental  frequency  of 
vibration.  To  obtain  approximate  values  of  the  frequencies  of  overtones  it 
is  necessary,  however,  to  use  a  series  of  several  terms  and  then  the  work 
becomes  laborious  as  it  is  necessary  to  solve  an  algebraic  equation  of  high 
order.  Many  other  methods  of  approximating  to  the  frequencies  of  over- 
tones are  now  available. 

Trefftz  has  recently  introduced  a  new  method  of  approximating  to  the 
solution  of  a  differential  equation,  in  which  the  original  variation  problem 
87  =  0  is  replaced  by  a  modified  variation  problem  87  (<r)  =  0  in  such  a  way 
that  the  desired  solution  u  can  be  expressed  in  the  form 


This  method,  combinfed  with  Trefftz's  method  of  estimating  the  error  of 
approximation  to  an  integral  such  as  /  (e),  can  lead  to  an  estimate  of  the 
error  involved  in  a  computation  of  u.  In  the  problem  of  the  deflection  of 
a  clamped  plate  under  a  given  distribution  of  load,  the  function  /  (e) 
represents  the  potential  energy  when  a  concentrated  load  €  is  placed  at  the 
point  where  the  deflection  u  is  required. 

Courant  has  shown  that  the  rapidity  of  convergence  of  a  method  of 
approximation  can  often  be  improved  by  modifying  the  variational 
problem,  introducing  higher  derivatives  in  such  a  way  that  the  Eulerian 
equatioA  of  the  problem  is  satisfied  whenever  the  original  differential 
equation  is  satisfied.  This  device  is  useful  also  in  applications  of  Trefftz's 
method. 

An  entirely  different  method  of  approximation  is  based  on  the  use  of 
difference  equations  which  in  the  limit  reduce  to  the  differential  equation 
of  a  problem.  The  early  writers  were  content  to  adopt  the  principle,  usually 
called  Rayleigh's  principle,  that  it  is  immaterial  whether  the  limiting  pro- 
cess is  applied  to  the  difference  equations  or  their  solutions.  Some  attempts 
have  been  made  recently  to  justify  this  principle*  and  also  to  justify  the 
use  of  a  similar  principle  in  the  treatment  of  problems  of  the  Calculus  of 
Variations  by  a  direct  method,  due  to  Euler,  in  which  an  integral  is  replaced 
by  a  finite  sum.  An  example  indicating  the  use  of  partial  difference 
equations  and  finite  sums  is  discussed  in  §  2-33. 

*  See  the  paper  by  N.  Bogoliouboff  and  N.  Kryloff,  Annals  of  Math.  vol.  xxrx,  p.  255  (1928). 
Many  references  to  the  literature  are  contained  in  this  paper.  In  particular  the  method  is  discussed 
by  R.  B.  Robbing,  Amer.  Journ.  vol.  xxxvn,  p.  367  (1915). 


CHAPTER  I 
THE   CLASSICAL  EQUATIONS 

§  1-11.  Uniform  motion.  It  seems  natural  to  commence  a  study  of  the 
differential  equations  of  mathematical  physics  with  a  discussion  of  the 
equation  ^ 

3*"°' 

which  is  the  equation  governing  the  motion  of  a  particle  which  moves  along 
a  straight  line  with  uniform  velocity.  It  may  be  thought  at  first  that  this 
equation  needs  no  discussion  because  the  general  solution  is  simply 

x  =  At  +  B, 

where  A  and  B  are  arbitrary  constants,  but  in  mathematical  physics  a 
differential  equation  is  almost  always  associated  with  certain  supple- 
mentary conditions,  and  it  is  this  association  which  presents  the  most 
interesting  problems. 

A  similar  differential  equation 


describes  an  essential  property  of  a  straight  line,  when  x  and  y  are  inter- 
preted as  rectangular  co-ordinates,  and  its  solution 

y  =  mx  +  c 

is  the  f amiliar  equation  of  a  straight  line :  the  property  in  question  is  that 
the  line  has  a  constant  direction,  the  direction  or  slope  of  the  line  being 
specified  by  the  constant  m.  For  some  purposes  it  is  convenient  to  regard 
the  line  as -a  ray  of  light,  especially  as  the  conditions  for  the  reflection  and 
refraction  of  rays  of  light  introduce  interesting  supplementary  or  boundary 
conditions,  and  there  is  the  associated  problem  of  geometrical  foci  of  a 
system  of  lenses  or  reflecting  surfaces. 

If  a  ray  starts  from  a  point  Q  on  the  axis  of  the  system  and  is  reflected 
or  refracted  at  the  different  surfaces  of  the  optical  system  it  will,  after 
completely  traversing  the  system,  be  transformed  into  a  second  ray  which 
meets  the  axis  of  the  system  in  a  point  Q,  which  is  called  the  geometrical 
focus  of  Q.  The  problem  is  to  find  the  condition  that  a  given  point  Q  may 
be  the  geometrical  focus  of  another  given  point  Q. 

This  problem  is  generally  treated  by  an  approximate <  method  which 
illustrates  very  clearly  the  mathematical  advantages  gained  by  means  of 
simplifying  assumptions.  It  is  assumed  that  the  angle  between  the  ray 
and  the  axis  is  at  all  times  small,  so  that  it  can  be  represented  at  any  time 
by  dy/dx. 


2  The  Classical  Equations 

Let  y2  '=  4a#  give  an  approximate  representation  of  a  refracting  surface 
in  the  immediate  neighbourhood  of  the  point  (0,  0)  on  the  axis.  If  y  is  a 
small  quantity  of  the  first  order  the  value  of  x  given  by  this  equation  can 
be  regarded  as  a  small  quantity  of  the  second  order  if  a  is  of  order  unity. 
Neglecting  quantities  of  the  second  order  we  may  regard  x  as  zero  and  may 
denote  the  slope  of  the  normal  at  (#,  y)  by 

_  dx  „  y 

dy     2a' 

Now  let  suffixes  1  and  2  refer  to  quantities  relating  to  the  two  sides  of 
the  refracting  surface.  Since  the  angle  between  a  ray  and  the  normal  to 
the  refracting  surface  is  approximately  dy/dx  -f-  y/2a,  the  law  of  refraction 
is  represented  by  the  equations 


Denoting  by  [u]  the  discontinuity  u2  —  u±  in  a  quantity  u,  we  have  the 
boundary  conditions 

[dy\ 


[y]  =  o. 

Dropping  the  suffixes  we  see  that  these  boundary  conditions  are  of  type 


[y]  =  o, 

where  A  and  B  are  constants  which  may  be  either  positive  or  negative. 

In  the  case  of  a  moving  particle,  which  for  the  moment  we  shall  regard 
as  a  billiard  ball,  a  supplementary  condition  is  needed  when  the  ball  strikes 
another  ball,  which  for  simplicity  is  supposed  to  be  moving  along  the  same 
line.  If  Ui  ,  u2  are  the  velocities  of  the  first  ball  before  and  after  collision, 
U19  U2  those  of  the  second  ball  before  and  after  collision,  the  laws  of 
impact  give  u>-  U,=  -  e  (u,  -  17,), 


mu2  +  MU2  = 

where  e  is  the  coefficient  of  restitution  and  m,  M  are  the  masses  of  the  two 
balls.  Regarding  Ul  as  known  and  eliminating  U2  we  have 

0  (M  -f-  m)  u2  =  (m  -  eM)  u±  +  M  (1  +  e)  Ul  . 

Replacing  u2  —  u±  by  [dx/dt]  we  have  the  boundary  conditions  for  the 
collision 

[x]  =0,  (  M  4-  m)  [g]  =  M  (  1  +  e)  (  U,  - 


Boundary  Conditions  3 

These  hold  for  .the  place  x  =  xl  where  the  collision  occurs,  x  being  the  co- 
ordinate of  the  centre  of  the  colliding  ball. 

The  boundary  conditions  considered  so  far  may  be  included  in  the 
general  conditions  , 

' 


[y]  =  o, 

where  A,  JB  and  C  are  constants  associated  with  the  particular  boundary 
under  consideration. 

§  1-12.  Other  types  of  boundary  condition  occur  in  the  theory  of  uni- 
form fields  of  force. 

A  field  of  force  is  said  to  be  uniform  when  the  vector  E  which  specifies 
the  field  strength  is  the  same  in  magnitude  and  direction  for  each  point 
of  a  certain  domain  D.  Taking  the  direction  to  be  that  of  the  axis  of  x  the 
field  strength  E  may  be  derived  from  a  potential  V  of  type  V  —  Ex  by 
means  of  the  equation  ,„ 

E  =  -  ,  > 
ax 

V  being  an  arbitrary  constant.  This  potential  V  satisfies  the  differential 
equation  /2  r/ 


throughout  the  domain  D. 

Boundary  conditions  of  various  types  are  suggested  by  physical  con- 
siderations. At  the  surface  of  a  conductor  V  may  have  an  assigned  value. 

At  a  charged  surface     -j-     may  have  an  assigned  value,  while  there  may 

be  a  surface  at  which  [F]  has  an  assigned  value  (contact  difference  of 
potential). 

With  boundary  conditions  of  the  types  that  have  already  been  con- 
sidered many  interesting  problems  may  be  formulated.  We  shall  consider 
only  two. 

§  1-13.  Problem  1.  To  find  a  solution  of  d2y/dx2  =  0  which  satisfies  the 
conditions 

y  =  0  when  x  =  0  and  when  x  =  1  ;    [dy/dx]  —  —  1  when  x  —  £  ; 

[y]  =  o- 

The  first  condition  is  satisfied  by  writing 

y  =  Ax  x<  g 

=  JS(l-a;)        x>£. 
The  condition  [y]  =  0  gives 

A£  =  B  (1  -  fl, 


4  The  Classical  Equations 

and  the  condition  [dy/dx]  =  —  1  gives 

A  +  B=\. 

.:   A=l-f,        B  =  f 
Hence  y  =  9  (x,  £)  =  a;  (1  -  £)         (a;  < 


This  function  is  called  the  Green's  function  for  the  differential  expression 
d2y/dx2,  on  account  of  its  analogy  to  a  function  used  by  George  Green  in  the 
theory  of  electrostatics. 

It  may  be  remarked  in  the  first  place  that  a  solution  of  type  P  +  Qx 
which  satisfies  the  conditions  y  ~  a  when  x  =  0,  y  =  b  when  x  =  1,  is  given 
by  the  formula 


Secondly,  it  will  be  noticed  that  </  (a;,  £)  is  a  symmetrical  function  of  x 
and  £;  in  other  words  g  (x,  £)  =  g  (£,  #)• 

A  third  property  is  obtained  by  considering  a  solution  of  d2y/dx2  —  0 
which  is  a  linear  combination  of  a  number  of  such  Green's  functions,  for 
example,  n 

y=   £  fsg  (x,  £,), 

s-l 

where  flt  /2,  /3,  ...  are  arbitrary  constants.  The  derivative  dy/dx  drops  by 
an  amount/j  at  gl9  by  an  amount  /2  at  £2,  and  so  on. 
*    Let  us  now  see  what  happens  when  we  increase  the  number  of  points 
f  i  ,  f  2  >  &  >  •  •  •  and  proceed  to  a  limit  so  that  the  sum  is  replaced  by  an  integral 

y=flog(x,$)f(€)d£  ......  (A) 


JO 

We  find  on  differentiating  that 


=  -  xf(x)  -  (1  -  x)f(x)  =  -/(«), 

the  function  /  (x)  being  supposed  to  be  continuous  in  the  interval  (0,  1). 
It  thus  appears  that  the  integral  is  no  longer  a  solution  of  the  differential 
equation  d^/dx2  =  0,  but  is  a  solution  of  the  non-homogeneous  equation 


Conversely,  if  the  function  /  (x)  is  continuous  in  the  interval  (0,  1)  a 
solution  of  this  differential  equation  and  the  boundary  conditions,  y  =  0 
when  x  =  0  and  when  x  =  1,  is  given  by  the  formula  (A);  this  formula, 


The  Green's  Function  5 

moreover,  represents  a  function  which  is  continuous  in  the  interval  and 
has  continuous  first  and  second  derivatives  in  the  interval.  Such  a  function 
will  be  said  to  be  continuous  (D,  2),  or  of  class  C"  (Bolza's  notation). 

§  1-14.   Problem  2.  To  find  a  solution  of  d2y/dx2  =  0  and  the  supple- 
mentary conditions 

[y]=°  U*-,M 

y  =  0  when  x  =  0  and  when  x  =  1  ;     n  [dyjdx]  +  k*y  =  0)  '    ' 

where  s  =  1,  2,  3,  ...  n  —  1. 

Let  y  =  ^48a;  -f  #8>         5  —  I  <  nx  <  s, 

then  the  supplementary  conditions  give  B±  =  0,  ^4n  -f  J5n  —  0, 


n 


?!  =  0,        7i  (A2  -  A,)  -f  £2  (AJn  -f  ^)    =  0, 


(A3  -  A,)  |  +  £3  -  J?2  =  0,        n  (A3  -  AJ  +  k*  (2A2/n  +  B2)  =  0, 

H 


nB3  =  (^!  +  ^42  -f  ...  A3)  -  sA8, 
n2  (AM  -  A8)  +  P  (Al  +  A2+  ...  ^f)  =  0, 

**  (A*+i  -  2^«  +  A*-i)  +  &AB  =0,  s  >  1. 

This  difference  equation  may  be  solved  by  writing  k2  =  2n2  (1  —  cos  6), 

.-.   ^8  =  A!  cos  («  -  1)  0  +  K  sin  (s  -  1)  0, 
where  K  is  a  constant  to  be  determined.   Now 

A2  =  Ai  +  2Al  (cos  6  -  1)  =  Al  (2  cos  0  -  1), 
therefore  K  sin  0  =  Al  (cos  0  —  1), 


_  .        A  sin  sQ  —  sin  5— 

and  so  As  =  Al  - 


The  condition  0  =  An  -f  Bn  is  satisfied  if  nO  =  r-n,  where  r  is  an  integer. 
In  the  limit  when  n  =  oo  this  condition  becomes 

k  =  lim  2n  sin  —  =  TTT, 


6  The  Classical  Equations 

and  this  is  exactly  the  condition  which  must  b'e  satisfied  in  order  that  the 
differential  equation 


may  possess  a  non-  trivial  solution  which  satisfies  the  boundary  conditions 
y  =  0  when  x  =  0  and  when  x  =  1.  The  general  solution  of  this  equation  is, 

in  fact,  ~         ,         r\    •     i 

y  —  "  cos  kx  -f  Q  sin  to, 

where  P  and  $  are  arbitrary  constants.  To  make  y  =  0  when  #  =  0  we 
choose  P  =  0.  The  condition  y  =  0  when  x  =  1  is  then  satisfied  with  Q  =£  0 
only  if  sin  &  ^  0,  i.e.  if  k  =  r?r. 

The  exceptional  values  of  P  of  type  (rTr)2  are  called  by  the  Germans 
"  Eigenwerte  "  of  the  differential  equation  (B)  and  the  prescribed  boundary 
conditions.  A  non-trivial  solution  Q  sin  (kx)  which  satisfies  the  boundary 
conditions  is  called  an  "Eigenfunktion."  These  words  are  now  being  used 
in  the  English  language  and  will  be  needed  frequently  in  this  book.  To 
save  printing  we  shall  make  use  of  the  abbreviation  eit  for  Eigenwert  and 
eif  for  Eigenfunktion.  The  conventional  English  equivalent  for  Eigenwert 
is  characteristic  or.  proper  value  and  for  Eigenfunktion  proper  function. 

The  theorem  which  has  just  been  discussed  tells  us  that  the  differential 
equation  (B)  and  the  prescribed  boundary  conditions  have  an  infinite 
number  of  real  eits  which  are  all  simple  inasmuch  as  there  is  only  one  type 
of  eif  for  each  eit.  The  eits  are,  moreover,  all  positive. 

The  quantities  &r2  =  (  2n  sin  --  j 

may  be  regarded  as  eits  of  the  differential  equation  d2y/dx*  =  0  and  the 
preceding  set  of  boundary  conditions.  These  eits  are  also  positive,  and  in 
the  limit  n  ->oo  they  tend  towards  eits  of  the  differential  equation  (B)  and 
the  associated  boundary  conditions.  The  solution  y  corresponding  to  k  is, 
for  s  —  1  <  nx  <  5, 


TT\  [f         S\  (  .    STTT         .     S  -  1  .  nrX       1    .     sr-rrl 
)\(x  —    -sin  ----  sin  --    -f  -  sin  —  ,  ......  (C) 

nl  [\        n>  \        n  n       J     n         n  \          v    ' 

and  it  is  interesting  to  study  the  behaviour  of  this  function  as  n  ->  oo  to  see 
if  the  function  tends  to  the  limit 

•*i-i    . 

f  ^si 

TTT 

T    ,  .,  A          A    fT7r\ 

Let  us  write  A0  =  A1(-~}  cosec 

0         1\nJ  \n 

A  A 

^o  (x)  =  r7r°  sin  (rirx),        Fl  (x)  =  —  x  sin  (mx), 

and  let  us  use  F  (x)  to  denote  the  function  (C)  which  represents  a  potygon 
with  straight  sides  inscribed  in  the  curve  y  —  FQ  (x). 


A  Passage  to  the  Limit  7 

The  closeness  of  the  approximation  of  F  (x)  to  F1  (x)  can  be  inferred 
from  the  uniform  continuity  of  FQ  (x). 

Given  any  small  quantity  6  we  can  find  a  number  n  (e)  such  that  for 
any  number  n  greater  than  n  (c)  we  have  the  inequality 


r.  (*)-*•('- 


<  € 


< 


for  any  point  x  in  the  interval  s  —  1  <  nx  <  s  and  for  any  value  of  s  in  the 
set  1,  2,  3,  ...  n.   In  particular 

-  1\       _  /s\ 
n    )         °  \n) 

Now    F  (x)  =  FQ  (-}  +  (s-  nx)  \FQ  (S  ~  l]  -  F0  (-][       (s-  Knx<  s). 
\n/  [_      \    n    /  \n/  j 

Therefore  |  F  (x)  -  FQ  (x)  \  <  e  +  c. 

On  the  other  hand 

IV    (/y\          V    I fy\   I   I    V    { rr\ 
•^0  \x)  ~  *  I  \X)   I   ~    I  ^0  \X) 

Therefore  <  A1\--  cosec 1 

IV 


|  ^  (a;)  -  ^  (a;)  |  <  2e  +  ^        cosec  ~~  - 

But  when  €  is  given  we  can  also  choose  a  number  m  (e)  such  that  for 
n  >  m  (e)  we  have  the  inequality 


A    [TTT  TTT       _  "1  I 

,    —  cosec  -----  1 

l[_n  n         J  I 


<  €. 


Consequently,  by  choosing  n  greater  than  the  greater  of  the  two 
quantities  n  (e)  and  m  (e),  if  they  are  not  equal,  we  shall  have 

\F(x)~  F,  (x)  |  <  36. 

This  inequality  shows  that  as  n  ->  oo,  F  (x)  tends  uniformly  to  the  limit 
F,  (x). 

This  method  of  obtaining  a  solution  of  the  equation 

2  +*•»-« 

from  a  solution  of  the  simpler  equation  d2y/dx2  =  0  by  a  limiting  process, 
can  be  extended  so  as  to'  give  solutions  of  other  differential  equations  and 
specified  boundary  conditions,  but  the  question  of  convergence  must  always 
be  carefully  considered. 

§  1*15.   Fourier's  theorem.  It  seems  very  nat.ural  to  try  to  find  a  solution 
of  the  equation 


and  a  prescribed  set  of  supplementary  conditions  by  expanding  /  (x)  in  a 


8  The  Classical  Equations 

d?ii 

series  of  solutions  of  -—£  +  k*y  =  0  and  the  prescribed  supplementary  con- 
ditions, because  if  f(x)=Xbnsinnx  (A) 

the  differential  equation  is  formally  satisfied  by  the  series 

~ ,    sin  nx 
y.S&.—i-, 

and  if  the  original  series  is  uniformly  convergent  the  two  differentiations 
term  by  term  of  the  last  series  can  be  justified.  When  the  f unction /(#) 
is  continuous  it  is  not  necessary  to  postulate  uniform  convergence  because 
Lusin  has  proved  that  if  the  series  (A)  converge  at  all  points  of  an  interval 
/  to  the  values  of  a  continuous  function/  (x)  then  the  series  (A)  is  integrable 
term  by  term  in  the  interval  7.  Unfortunately  it  has  not  been  proved  that 
an  arbitrary  continuous  function  can  be  expanded  in  a  trigonometrical 
series.  Indeed,  we  are  faced  with  the  question  of  the  possibility  of  expanding 
a  given  function  /  (x)  in  a  trigonometrical  series  of  type  (A).  This  question 
is  usually  made  more  definite  by  stipulating  the  range  of  values  of  x  for 
which  the  representation  of  /  (x)  is  required  and  the  type  of  function/  (x) 
to  which  the  discussion  will  be  limited.  A  mathematician  who  starts  out 
to  find  an  expansion  theorem  for  a  perfectly  arbitrary  function  will  find 
after  mature  consideration  that  the  programme  is  too  ambitious*,  as  there 
are  functions  with  very  peculiar  properties  which  make  trouble  for  the 
mathematician  who  seeks  complete  generality.  It  is  astonishing,  however, 
that  a  function  represented  by  a  trigonometrical  series  is  not  of  an  exceed- 
ingly restricted  type  but  has  a  wide  degree  of  generality,  and  after  the 
discussions  of  the  subject  by  the  great  mathematicians  of  the  eighteenth 
century  it  came  as  a  great  surprise  when  Fourier  pointed  out  that  a 
trigonometrical  series  could  represent  a  function  with  a  discontinuous 
derivative,  and  even  a  discontinuous  function  if  a  certain  convention  were 
adopted  with  regard  to  the  value  at  a  point  of  discontinuity.  In  Fourier's 
work  the  coefficients  were  derived  by  a  certain  rule  now  called  Fourier's 
rule,  though  indications  of  it  are  to  be  found  in  the  writings  of  Clairaut, 
Euler  and  d'Alembert.  In  the  case  of  the  sine-series  the  rule  is  that 


2  [w 
bn  =  -      /  (x)  sin  nxdx, 

it  Jo 


and  the  range  in  which  the  representation  is  required  is  that  of  the  interval 
(0  <  x  <  TT).  When  the  range  is  (0  <  x  <  2ir)  and  the  complete  trigono- 
metrical series  *  « 

/  (x)  =  $0o  +   2  an  cos  nx 

n-l 

00 

+    S  bnsinnx 

n-l 

*  For  the  history  of  the  subject  see  Hobson's  Theory  of  Functions  of  a  Real  Variable  and  Burk- 
hardt's  Report,  Jahresbericht  der  Deuteehen  Math.  Verein,  vol.  x  (1908). 


Fourier  Constants  9 

is  to  be  used  for  the  representation,  Fourier's  rule  takes  the  form 

an  =  -       /  (x)  cos  nxdx, 

TT  JQ 
1   f2tr 

bn  =  -      /  (x)  sin  nx  dx,  (B) 

TT  Jo 

and  the  coefficients  an ,  bn  are  called  the  Fourier  constants  of  the  function 


Unless  otherwise  stated  the  symbol/  (x)  will  be  used  to  denote  a  function 
which  is  single-valued  and  bounded  in  the  interval  (0,  2n)  and  defined  out- 
side this  interval  by  the  equation  /  (x  -f  277)  =  /  (x). 

For  some  purposes  it  is  more  convenient  to  use  the  range  (—  TT  <  u  <  TT) 
and  the  variable  u  =  2?r  —  x.  If  /  (x)  =  F  (u)  the  coefficients  in  the  ex- 
pansion of  F  (u)  in  a  trigonometrical  series  of  Fourier's  type  are  given  by 
formulae  exactly  analogous  to  (B)  except  that  the  limits  are  —  TT  and  TT 
instead  of  0  and  2?r. 

The  advantage  of  using  the  interval  (—  TT,  TT)  instead  of  the  interval 
(0,  277-)  is  that  if  F  (u)  is  an  odd  function  of  u,  i.e.  if  F  (—  u)  =  —  F  (u),  the 
coefficients  an  are  all  zero,  and  if  F  (u)  is  an  even  function  of  u,  i.e.  if 
F  (—  u)  —  F  (u),  the  coefficients  bn  are  all  zero.  In  one  case  the  series 
becomes  a  sine-series  and  in  the  other  case  a  cosine-series. 

The  possibility  of  the  expansion  of  /  (x)  in  a  Fourier  series  is  usually 
established  for  a  function  of  limited  variation*,  that  is  a  function  such  that 
the  sum  n_1 

s  I  /(*.«)-/(*.)  I 

s~0 

is  bounded  and  <  N9  say,  for  all  sets  of  points  of  subdivision  xl9x2y  ...  a?n-1 
dividing  the  interval  (0,  2n)  up  into  n  parts  and  for  all  finite  integral  values 
of  n.  Such  a  function  is  also  called  a  function  of  limited  total  fluctuation 
and  a  function  of  bounded  variation. 

In  addition  to  this  restriction  on  /  (x)  it  is  also  supposed  that  the 
integrals  in  the  expressions  for  the  coefficients  exist  in  the  ordinary  sensef. 
In  the  case  when  the  integral  representing  an  is  an  improper  integral  it  is 
assumed  that  the  integral 

2  (C) 


is  convergent.  If  x  is  any  interior  point  of  the  interval  (0,  27r)  it  can  be 
shown  that  when  the  foregoing  conditions  are  satisfied  the  series  is  con- 
vergent and  its  sum  is  c 

Mm 


*  Whittaker  and  Watson's  Modem  Analysis,  3rd  ed.  p.  175. 

f  That  is,  in  the  Riemann  sense.  There  are  corresponding  theorems  for  the  cases  in  which  other 
definitions  of  integral  (such  as  those  of  Stieltjes  and  Lebesgue)  are  used. 


10  The  Classical  Equations 

when  the  limits  oif(x±  e)  exist,  i.e.  with  a  convenient  notation 
H/(*  +  0)  +/(s  -  0)]  =/(x),  say. 

When  the  function  /  (x)  is  continuous  in  an  interval  (a  <  x  <  /?)  con- 
tained in  the  interval  (0,  2n),  is  of  limited  variation  in  the  last  interval  and 
the  other  conditions  relating  to  the  coefficients  are  satisfied,  it  can  be 
shown  *  that  the  series  is  uniformly  convergent  for  all  values  of  x  for  which 
«  -f  S  <  #  -'  /J  —  8,  where  S  is  any  positive  number  independent  of  x. 

When  the  conditions  of  continuity  and  limited  variation  are  dropped 
and  the  function  /  (x)  is  subject  only  to  the  conditions  relating  to  the 
existence  of  the  integrals  in  the  formulae  for  the  coefficients  and  the 
convergence  of  the  integral  (C),  there  is  a  theorem  due  to  Fejcr,  which 
states  thatf 

/>)  =  lim  i  {A0  +  S,  (x)  +  S2  (x)  +  ...  S,K_,  (x)}, 

m*-±  oo  •ft' 

in 

where     A0  =  |a0,     An  (x)  =  an  cos  nx  +  bn  sin  nx,     Sm(x)=   2  An  (x). 

«-<) 

This  means  that  the  series  is  summable  in  the  Cesaro  sense  by  the  simple 
method  of  averaging  which  is  usually  denoted  by  the  symbol  (C,  1). 

This  is  a  theorem  of  great  generality  which  can  be  used  in  applied 
mathematics  in  place  of  Fourier's  theorem.  It  is  assumed,  of  course,  that 
the  limits 

lim  /(a?+€)=/(a?  +  0),     Km  /(*?-€)=/(*-  0) 

t->  0  «->  0 

exist  J. 

§  1-16.   Cesaro's  metJwd  of  summation  §.   Let 


n8n  =  sl  +  82+  ...  +  sn, 
then,  if  Sn  ->  S  as  n  ->  oo,  the  infinite  series 


(1) 


is  said  to  be  summable  (C,  1)  with  a  Cesaro  sum  S. 

For  consistency  of  the  definition  of  a  sum  it  must  be  shown  that  when 
the  series  (1)  converges  to  a  sum  s,  we  have  8  =  S.  To  do  this  we  choose 
a  positive  integer  n,  such  that 

I  Sn+p-*n  |   <.€  ......  (2) 

for  all  positive  integral  values  of  p.  This  is  certainly  possible  when  s  exists 
and  we  have  in  the  limit 

|*-*n|   <*.  ......  (3) 

*  Whittaker  and  Watson,  Modern  Analysis,  3rd  ed.  p.  179. 

t  Ibid.  p.  169. 

J  When/  (a:  -f  0)  =  /  (x  -  0)  this  implies  that  f(x)  is  continuous  at  the  point  x. 

§  Bull,  des  Sciences  Math.  (2),  t,  xrsr,  p.  114  (1890).   See  also  Bromwich's  Infinite  Series. 


Fejer's  Theorem  11 

Now  let  v  be  an  integer  greater  than  n  and  let  Cm  be  defined  by  the 

equation  ~ 

vCm+1  =  v-m,  ......  (4) 

then  Sv  =  c^Ui  4-  c2^2  4-  ...  4-  cvuv.  ......  (5) 

But  Cj  >  c2  >  ...  >  cv  >  0,  hence  it  follows  from  (2)  that 

I  cn+l^n+l  4-  Cn-f2^n+2  ~H   •••   4"  CVUV  \   <  €Cn4.1, 

i-e.  I  S,  -  (q^  4-  C2u2  -f  ...  4-  cn^n)  |  <  ecn+1. 

Making  v  ->  oo  we  see  that  if  S  be  any  limit  of  $„ 

|S-a»|  <€.  ......  (6) 

Combining  (3)  and  (6)  we  find  that 

|  S  -  s  |  <  2e. 

Since  e  is  an  arbitrary  small  positive  quantity  it  follows  that  S  =  s  and 
so  the  sequence  $„  has  only  one  limit  s. 

§  1-17.  Fejer's  theorem.  Let  us  now  write  u^  =  ^40,  ?/n+1  =  ^4n  (x),  then, 
by  using  the  expressions  for  the  cosines  as  sums  of  exponentials,  it  is  readily 
found  that*  , 


where  26  =  |  a;  —  ^  |  .    Now  the  integrand  is  a  periodic  function  of  /  of 
jjb^ioil  277,  Consequently  we  may  also  write 


b^arthermore,  since 

«lr'^  =  /w  4.  2  (w  -  1)  cos  26  +  2  (m  -  2)  cos  46  +  .  .  .  +  2  cos  2  (m  -  1)  9 
sm'20 

•i  •          i-i  xu  A  f|7r  sin2  w0       | 

it  fs  readily  seen  that  ---.-  -9  «  =  ^TT. 

J  Jo  msm2^ 

I     Writing  ^  (0)  =  /  (x  4-  20)  +  /  (*  -  20)  -  2/  (a;), 

and  inaking  use  of  the  last  equation,  we  find  that 


w  ier«  <k  (0)  ->  0  as  0  ->  '0. 

How  if  e  is  any  small  positive  quantity  we  can  choose  a  number  8 


The  details  of  the  analysis  are  given  in  Whittaker  and  Watson's  Modern  Analysis 


12  The  Classical  Equations 

whenever  0  <  9  <  8,  and  if  €  is  independent  of  m  the  number  8  may  be 
regarded  as  independent  of  m. 
Writing  for  brevity 

sin2  m6  =  m  sin2  9P  (0),     TT  -  2« 
and  noting  that  P  (6)  is  never  negative,  we  have 


o 


(9)  fa  (9)  d9 


P  P  (9)  |  +x  (9)  |  d9  +  p  P  (9)  \  <t>x  (9)  \  d9 
Jo  Js 


< 

Let  us  now  suppose  that         |  /  (t)  \  dt  exists,  then 

J   —  TT 

f  I  &  (9)  I  d9 
Jo 

also  exists,  and  by  choosing  a  sufficiently  large  value  of  m  we  can  make 

aem  sin2  8  >  f"  I  J>  (9)  I  d9. 
Jo 

This  makes  the  second  integral  on  the  right  of  (B)  less  than  «e,  which 
is  also  the  value  of  the  first  integral.  Therefore 

\Sm(x)-f(x)\  <2ae/7r=e; 
consequently  Sm  (x)  ->  /  (x)  as  m  ->  oo. 

When/  (x)  is  continuous  throughout  the  interval  (—  TT  <  x  <  TT)  all!  ,i 
foregoing  requirements  are  satisfied  and  in  addition  /  (x)  =  f  (x) ;  col  & 
quently,  in  this  case,  Sn  (x)  ->  f  (x),  and  this  is  true  for  each  point  I  o 
the  interval.  \  f 

This  celebrated  theorem  was  discovered  by  Fejer*.  The  conditional  oJ 
the  theorem  are  certainly  satisfied  when  the  range  (—  TT  <  x  <  TT)  canpbe 
divided  up  into  a  finite  number  of  parts  in  each  of  which  /  (x)  is  bo <mded 
and  continuous.  Such  a  function  is  said  to  be  continuous  bit  by  bit 
(Stiickweise  stetig);  the  Cesaro  sum  for  the  Fourier  series  is  then/ (x)  at 
any  point  of  the  range,  /  (x)  and  /  (x)  being  the  same  except  at  the  points 
of  subdivision. 

§  1-18.  ParsevaTs  theorem.   Let  the  function  /  (x)  be  continuous  bit  by 
bit  in  the  interval  (—  TT,  ?r)  and  let  its  Fourier  constants  be  an,  6n;  it 
then  be  shown  that 


[/  (*)?  dx  =  TT  [~K2  +  2  (an2  -f  6n2)l  .......  (A) 

L  n=l  J 


Math.  Ann.  Bd.  LVIII,  S.  51  (1904). 


ParsevaVs  Theorem 


13 


We  shall  find  it  convenient  to  sum  the  series  (A)  by  the  Ces&ro  method* 
This  will  give  the  correct  value  for  the  sum  because  the  inequality 


;-f 


n-l 


n-1 


indicates  that  the  series  is  convergent. 

To  find  the  sum  (C,  1)  we  have  to  find  the  limit  of  Sm  where,  by  a  simple 
extension  of  1-17  (A), 


S  -- 

M  ~  277    _      .^ 


sn* 


29  being  equal  to  |  x  —  t  \  . 

Since  the  region  of  integration  can  be  divided  up  into  a  finite  number 
of  parts  in  each  of  which  the  integrand  is  a  continuous  function  of  x  and  t, 
the  double  integral  exists  and  can  be  transformed  into  a  repeated  integral 
in  which  x  and  0  are  the  new  independent  variables.  The  region  for  which 
6  lies  between  6Q  and  00  -f  dd,  while  x  lies  between  XQ  and  XQ  -h  dx  consists 
of  two  equal  partsf;  sometimes  two,  sometimes  one  and  sometimes  none 
of  these  parts  lie  within  the  region  of  integration.  When  this  is  taken 
into  consideration  the  correct  formula  for  the  transformation  of  the 
integral  is  found  to  be 


1       fir 

.=  H- 

*TTJQ 


n-20 


20) 


fTr 
JW-T 


In  the  derivation  of  this  result  Fig.  1  will 
be  found  to  be  helpful.  The  lines  MlM2y 
MZM±  are  those  on  which  0  has  an  assigned 
value,  while  N1N2,  JV3A74  are  lines  on  which  0 
has  a  different  assigned  value.  It  will  be 
noticed  that  a  line  parallel  to  the  axis  of  t 
meets  M1M2 ,  M3M^  either  once  or  twice,  while 
it  meets  N1N29  N^N^  either  once  or  not  at  all. 

Applying  the  theorem  of  §  1-17  to  (B)  we 
get 


-7T 


......  (B) 


7T-20 


7T 


Fig.  1. 


-  r  f(x)/(x)dx, 

J  —  IT 

and  when/ (x)  is  defined  to  be/ (x)  this  result  gives  (A). 


lim  , 

m->oo 


*  This  is  the  plan  adopted  in  Whittaker  and  Watson's  Modern  Analysis,  p.  181.  The  present 
proof,  however,  differs  from  that  given  in  Modern  Analysis,  which  is  for  the  case  in  which  f  (x)  is 
bounded  and  integrable. 

t  It  will  be  noted  that  the  Jacobian  of  the  transformation  has  a  modulus  equal  to  two. 


14  The  Classical  Equations 

The  theorem  (A)  was  first  proved  by  Liapounoff*;  the  present  investi- 
gation is  a  modification  of  that  given  by  Hurwitzf. 

Now  let  F  (x)  be  a  second  function  which  is  continuous  bit  by  bit  in 
the  interval  —  -n  <  x  <  77  and  let  An ,  Bn  be  its  Fourier  constants.  Applying 
the  foregoing  theorem  to  F  (x)  -f  /  (x)  and  F  (x)  —  f  (x),  we  obtain 

T   [F  (x)  +  f  (x)Y  dx  =  77  [i  (A0  +  a0)2  -f  S  {(4n  +  aj2  -f 

J-7T  L  W=l 

I"   [F  (x)  -  f  (x)]*  dx  -  TT  [4  Mo  -  a0)2  +  S  {(4n  -  a  J2  + 

./-*  L  n-1  / 

Subtracting,  we  obtain  the  important  formula 

I*    /  (x)  F  (x)  dx  =  >n  \!>A0a0  +  2  (,4nan  +  Bnbn) 

J-1T  L  W-l 

which  is  usually  called  Parseval's  theorem,  though  ParsevaFs  derivation 
of  the  formula  was  to  some  extent  unsatisfactory. 

In  the  modern  theory,  when  Lebesgue  integrals  are  used,  the  theorem 
is  usually  established  for  the  case  in  which  the  functions  f  (x),  F  (x), 
[f  (x)]2  and  [F  (x)]2  are  integrable  in  the  sense  of  Lebesgue.  There  is  also 
a  converse  theorem  which  states  that  when  the  series  (A)  converges  there 
is  a  function  /  (x)  with  an  and  6n  as  Fourier  constants  which  is  such  that 
[/  (#)]2  is  integrable  and  equal  to  the  sum  of  the  series.  This  theorem  was 
first  proved  by  Riesz  and  Fischer.  Several  proofs  of  the  theorem  are  given 
in  a  paper  by  W.  H.  Young  and  Grace  Chisholm  Young  J.  The  theorem  has 
also  been  extended  by  W.  H.  Young §,  the  complete  theorem  being  also 
an?> extension  of  Parseval's  theorem.  A  general  form  of  Parseval's  theorem 
has  been  used  to  justify  the  integration  term  by  term  of  the  product  of  a 
function  and  a  Fourier  series. 

ADDITIONAL  RESULTS 

1.  If  the  functions  /  (x),  F  (x)  are  integrable  in  the  sense  of  Lebesgue,  and  [/(z)]2, 
[F  (x)]2  are  also  integrable  in  the  same  sense,  then|| 


T 

W   J    -T 


x)F(t)dt  =  K^o+    2    (anAn  H-  bnBn)  cos  nx  -    2    (anBn  -  bnAn)  sin  nx. 


2.  If  /  (x)  is  a  periodic  function  of  period  2?r  which  is  integrable  in  the  sense  of  Lebesgue, 
and  if  g  (x)  is  a  function  of  bounded  variation  which  is  such  that  the  integral 

'00 

\g(x)\  dx 
o 

*  Comptes  Rendus,  t.  cxxvi,  p.  1024  (1898). 

t  Math.  Ann.  Bd.  LVII,  S.  429  (1903). 

J  Quarterly  Journal,  vol.  XLJVI  p.  49  (1913). 

^  Comptes  Rendus,  t.  CLV,  pp.  30,  472  (1912);  Proc.  Roy.  Soc.  London,  A,  vol.  LXXXVII,  p.  331 
(1912);  Proc.  London  Math.  Soc.  (2),  vol.  xii,  p.  71  (1912).  See  also  F.  Hausdorff,  Math.  Zeits. 
Bd.  xvi,  S.  163(1923). 

||  W.  H.  Young,  Comptes  Rendus,  t.  CLV,  p.  30  (1912);  Proc.  Roy.  Soc.  London,  A,  vol. 
LXXXVII,  p.  331  (1912). 


Fourier  Series  15 

is  convergent,  then  the  value  of  the  integral 

r/(x)g(x)dx 
Jo 

may  be  calculated  by  replacing  /  (x)  by  its  Fourier  series  and  integrating  formally  term  by 
term.  In  particular,  the  theorem  is  true  for  a  positive  function  g  (x)  which  decreases  steadily 
as  x  increases  and  is  such  that  the  first  integral  is  convergent. 

[W.  H.  Young,  Proc.  London  Math.  Soc.  (2),  vol.  ix,  pp.  449,  463  (1910);  vol.  xin,  p.  109 
(1913);  Proc.  Roy^oc.  A,  vol.  xxxv,  p.  14  (1911).  G.  H.  Hardy,  Mess,  of  Math.  vol.  LI, 
p.  186(1922).] 

§  1-19.  The  expansion  of  the  integral  of  a  bounded  function  which  is 
continuous  bit  by  bit.  If  in  Parse  val's  theorem  we  put 

F  (x)  =  1  ,     -  TT  <  x  <  z,     F  (x)  =  0,     z  <  x  <  TT, 
we  have  A^  =  -        F  (x)  dx  =  -        dx  =    -    n  , 

TT  J  -n  7T  J-TT  TT 

An  =  -          cos  nx.dx  =        [sin  nz], 
TT  )  -n  n-n 

Bn  =  -         sin  nx.dx  —  —  [cos  nn  —  cos  nz], 

TT  j  _ff  nrr 

and  we  have  the  result  that 

JZ  co        ] 

/  (x)  dx  =  |a0  (z  -f  77)  -|-    2    -  [an  sin  nz  -\-  bn  (cos  WTT  —  cos  nz)]. 
-TT  n=-i  n 

......  (A) 

Now  the  function  ^a0z  can  be  expanded  in  the  Fourier  series 

-    1 

—  «o  2   ~  cos  mr  sin  /i^, 
«=  i 

hence  the  integral,  on  the  left  of  (A)  can  be  expanded  in  a  convergent 
trigonometrical  series.  To  show  that  this  is  the  Fourier  series  of  the 
function  we  must  calculate  the  Fourier  constants. 

n 

dx 


.  .  an     a() 

H  —         —  /  (2)  c^s  nz  =  -----    cos  mr, 
J  v  '  n       n 


1  [n  [z  1  (n 

Now     -        sin  nz  dz        f  (x)  dx  =  ----  cos  mr        f  (x) 

7Tj-n  J—rr  M™  J  -n 

If71"    dz  p  .  . 

—  /  (2) 

TT  ]-„  n  J  v  ' 

1  f*  [s  1    [" 

cos  nzdz        f  (x)  dx  =  —  dz  sin  nz  f  (z) 

TTJ-rr  J-n  HIT  J  -n 

-^- 

-  f    dz  f    /  (x)  dx=\*    f  (x)  dx  -  !  f    zf(z)  dz 

TT)-*  J-n  J  -n  ^  J  -n 


OJ          J 

TTO,,  +  2    -  &„  cos  n?r, 


16  The  Classical  Equations 

by  Parseval's  theorem.  Hence  the  coefficients  are  precisely  the  Fourier 
constants  and  so  the  integral  of  a  function  which  is  continuous  bit  by  bit 
can  be  expanded  in  a  Fourier  series.  This  means  that  a  continuous  periodic 
function  with  a  derivative  continuous  bit  by  bit  can  be  expanded  in  a 
Fourier  series. 

Proofs  of  this  theorem  differing  from  that  in  the  text  are  given  by 
Hilbert-Courant,  Methoden  der  Mathematischen  Physik,  Bd.  I  (1924),  and 
by  M.  G.  Carman,  Bull.  Amer.  Math.  Soc.  vol.  xxx,  p.  410  (1924). 

It  should  be  noticed  that  equation  (A)  shows  that  when  /  (x)  can  be 
expanded  in  a  Fourier  series  this  series  can  be  integrated  term  by  term. 
A  more  general  theorem  of  this  type  is  proved  by  E.  W.  Hobson,  Journ. 
London  Math.  Soc.  vol.  n,  p.  164  (1927). 

Fourier's  theorem  may  be  extended  to  functions  which  become  infinite 
in  certain  ways  in  the  interval  (0,  277).  When  the  number  of  singularities  is 
limited  the  singularities  may  be  removed  one  by  one  by  subtracting  from 
/  (x)  a  simple  function  hs  (x)  with  a  singularity  of  the  same  type.  This 
process  is  continued  until  we  arrive  at  a  function 

0  (*)=/(*)-   2  h.(x) 

S-l 

which  does  not  become  infinite  in  the  interval  (0,  277).  The  problem  then 
reduces  to  the  discussion  of  the  Fourier  series  associated  with  each  of  the 
functions  hs  (x). 

§  1-21.  The  bending  of  a  beam.  We  shall  now  consider  some  boundary 
problems  for  the  differential  equation  d*yfdx*  =  0,  which  is  the  natural  one 
to  consider  after  d*y/dx2  =  0  from  the  historical  standpoint  and  on  account 
of  the  variety  of  boundary  conditions  suggested  by  mechanical  problems. 

The  quantity  y  will  be  regarded  here  as  the  deflection  from  the  equi- 
librium position  of  the  central  axis  of  a  long  beam  at  a  point  Q  whose 
distance  from  one  end  is  x.  The  beam  will  be  assumed  to  have  the  same 
cross-section  at  all  points  of  its  length  and  to  be  of  uniform  material,  also 
the  deflection  at  each  point  will  be  regarded  as  small.  The  physical  pro- 
perties of  the  beam  needed  for  the  simple  theory  of  flexure  are  then 
represented  simply  by  the  value  of  a  certain  quantity  B  which  is  called 
the  flexural  rigidity  and  which  may  be  calculated  when  the  form  of  the 
cross-section  and  the  elasticity  of  the  material  of  the  beam  are  known. 
We  are  not  interested  at  this  stage  in  the  calculation  of  B  and  shall  conse- 
quently assume  that  the  value  of  B  for  a  given  beam  is  known.  The  funda- 
mental hypothesis  on  which,  the  theory  is  based  is  that  when  the  beam  is 
bent  by  external  forces  there  is  at  each  point  x  of  the  central  axis  a  resisting 
couple  proportional  to  the  curvature  of  the  beam  which  just  balances  the 
bending  moment  introduced  by  the  external  forces.  When  the  flexure 
takes  place  in  the  plane  of  xy  this  resisting  couple  has  a  moment  which 


S+dS 


Bending  of  a  Beam  17 

can  be  set  equal  to  JBd*y/dx2  and  the  fundamental  equation  for  the  bending 
moment  is  M  =  Bdty(da.tm 

The  origin  of  the  bending  moment  will 
be  better  understood  when  it  is  remarked 
that  the  bending  moment  M  is  associated 
with  a  transverse  shearing  force  8  by  the 
equation 

-  s  =  dM/dx. 

When  the  beam  is   so   light  that  its  Fig.  2. 

weight  may  be  disregarded,  this  shearing  force  S  is  constant  along  any 
portion  of  the  beam  that  does  not  contain  a  point  of  support  or  point  of 
attachment  of  a  weight.  If  we  have  a  simple  cantilever  OA  built  into  a 
wall  at  O  and  carrying  a  weight  W  at  the  point  B  the  shearing  force  8  is 
zero  from  A  to  B  and  is  W  from  B  to  0, 
while  M  is  zero  from  A  to  B  and  equal 
to  Wx  between  B  and  0.  At  the  point 
0  the  fact  that  the  beam  is  built  in  or 
clamped  implies  that  ?/==  0  and  dy/dx  =  0, 
consequently  the  equation 

Bd*y/dx*  =  -  W  x  +  W  b 
gives    y  =  -  W  x*/6B  +  Wbx2/2B. 


O 


B 


W 


-a- 


This  holds  for  x  <  b.   For  x  >  b  the  differential  equation  for  y  is 

Bd*y/dx*  =  0, 
and  so  y  —  mx  +  c. 

The  quantities  y  and  dy/dx  are  supposed  to  be  continuous  at  B  and  so 
we  have  the  equations 

mb  +  c  =  W  b*/3B,        m  =  PF62/25 

which  give  c  ==  —  Wb3/6B.  The  deflection  of  B  is  Wb3/3B  and  is  seen  to  be 
proportional  to  the  force  W,  The  deflection  of  A  is  also  proportional  to  W. 

§  1-22.  Let  us  next  consider  the  deflection  of  a  beam  of  length  I  which 
is  clamped  at  both  ends  x  =  0,  x  =  I  and  which  carries  a  concentrated 
load  W  at  the  point  x  =  £. 

We  have  the  equations 


S  =  I7  +  ff,  S  =  T,  say, 

-  M  =  (T+  W"  )x  +  ^=  -Bd2y/dx*,    -  M  =  Tx 
£  (T7  +  IF)  x2  +  NX  -  -  JWy/ete,      ^T7^2 


$(T+W)x*+  $Nx*  =  -  By, 


W£  +  N  =  -  Bd2y/dx2, 

N)(x-l)  =  -  Bdy/dx, 
(a:  -  /) 


18  The  Classical  Equations 

where  T  and  N  are  constants  to  be  determined.  M  has  been  made  con- 
tinuous at  x  =  £,  but  we  have  still  to  make  y  and  dy/dx  continuous.  This 
gives  the  equations 

\(W&-  T/2)-  Wlg  +  Nl, 
i  (Wf»  +  TZ»)  =  j  (_  Pf  +  T)  Pf  +  prj  (2f  _  i)  + 

Therefore       Tl*  =  W£*  (2f  -  3Z),         ZW  =  Iff  (2£Z  -  Z2  - 

(Z  -  £)2  [a;  (Z  +  2f) 


This  solution  will  be  written  in  the  form  By  =  —  TFgr  (#,  f  )  and  the 
function  gr  (x,  ^)  will  be  calle.d  a  Green's  function  for  the  differential  ex- 
pression d*yldx*  and  the  prescribed  boundary  conditions. 


If  By= 

it  is  found  on  differentiation  that  y  is  a  solution  of  the  differential  equation 


the  function  w  (x)  being  supposed  to  be  continuous  in  the  range  (0,  /). 
Thi^  solution  corresponds  to  the  case  of  a  distributed  load  of  amount  wdx 
for  a  length  dx.  -When  w  is  independent  of  x  the  expression  found  for  y  is 

in 

By=^x*(l-x)*. 

It  should  be  noticed  that  the  Green's  function  g  (x,  f  )  is  a  symmetrical 
function  of  x  and  y,  its  first  two  derivatives  are  continuous  at  x  =  f  ,  but 
the  third  derivative  is  discontinuous,  in  fact 


The  reactions  at  the  ends  of  the  clamped  beam  with  concentrated  load 
are  found  by  calculating  the  shear  S.  When  x  <  £  we  have 

S  =  W  (I  -  ^)2  (Z  +  2f  )/Z», 

and  this  is  equal  in  magnitude  to  the  reaction  at  x  ==  0.  The  reaction  at 
x  =  I  is  similarly  R  =  W  (31  -  2f)  f  2/Z'. 

The  deflection  of  the  point  x  =  f  is 

%  =  W?  (I  -  WIWB, 

when  £  -  Z/2  this  amounts.to  W7S/192J3. 

In  the  case  of  the  uniformly  loaded  beam  the  reactions  at  the  ends  are 
respectively  £  W  and  |  W  as  we  should  expect.  The  deflection  of  the  middle 
point  is  W  Z4/384£. 


The  Green's  Functions  19 

§  1*23.    When  the  beam  is  pin -jointed  at  both  ends,  M  is  zero  there  and 
the  boundary  conditions  are 

y  =  0,     d2y/dx2  =  0  for  x  =  0  and  x  =  a. 

When  there  is  a  concentrated  load  IF  at  x  =  £  and  the  beam  is  of 
negligible  weight,  the  solution  is  By  —  Wk  (x,  £),  where 
&  (x,  f)  =  x  (a  -  £)  (x2  +  £2  - 
=  g(a-  x)  (x2  -f  £2  - 
The  reactions  at  the  supports  are 

J?0  =  W  (1  -  |/a)     at  a  =  0, 
J^a  =  W^/fl  at  a:  =  a. 

As  before,  the  deflection  corresponding  to  a  distributed  load  of  density 
w  (x)  is 


and  when  w  is  constant 

By  =  w  (x*  -  2ax*  -f  a3#)/24. 
The  reactions  at  the  supports  are  in  this  case 

7?  =  wcl  —  n 

./TO      "  o  a  u  x  — •  v/j 

2i 

T>  W(l 

jKa  —  —         at*  x  ===  a* 

In  the  case  of  a  beam  of  length  Z  clamped  at  the  end  x  =  0  and  pin- 
jointed  at  the  end  x  =  Z,  the  solution  for  the  case  of  a  concentrated  load 
W  at  x  =  |  is 

The  deflection  at  #  =  f  is  now 

y  =  Tf£3  (j  _  ^2  (4jj  _  |)/12£ 

when  |  =  Z/2,  and  the  reaction  at  #  =  Z  is 

If,  on  the  other  hand,  we  consider  a  beam  which  is  clamped  at  the  end 
x  =  0  but  is  free  at  the  end  x  =  Z  except  for  a  concentrated  load  P  which 
acts  there,  we  have,  at  the  point  x  —  £, 

while  at  the  point  x  —  I 


Byl  = 
Hence  R  =  Wyf:jyl . 

If  the  original  beam  is  acted  on  by  a  number  of  loads  of  type  W  we 
have,  for  the  reaction  at  the  end  x  =  Z, 


20  The  Classical  Equations 

On  account  of  this  relation  the  curve  (I)  is  called-the  " influence  line" 
of  the  original  beam.  Much  use  is  made  now  of  influence  lines  in  the  theory 
of  structures*. 

There  are  three  reciprocal  theorems  analogous  to  g  (x,  £)  =  g  (£,  x) 
which  are  fundamental  in  the  theory  of  influence  lines.  These  theorems, 
which  are  due  to  Maxwell  and  Lord  Rayleigh,  may  be  stated  as  follows: 

Consider  any  elastic  structure  with  ends  fixed  or  hinged,  or  with  one 
end  fixed  and  the  other  hinged,  to  an  immovable  support,  then 

(1)  The  displacement  at  any  point  A  due  to  a  load  P  applied  at  any 
point  B  is  equal  to  the  displacement  at  B  due  to  the  same  load  P  placed 
at  A  instead  of  B. 

(2)  If  the  displacement  at  any  point  A  is  prevented  by, a  load  P  at 
A  with  displacement  yB  at  B  under  a  load  Q,  and  alternatively  if  a  load 
Ql  at  B  prevents  displacement  at  B  with  displacement  yA  at  A  under  a 
load  P,  then  if  yA  =  yR ,  P  must  equal  Ql . 

(3)  If  a  force  Q  acts  at  any  point  B  producing  displacements  yB  at 
B  and  yA  at  any  other  point  A,  and  if  a  second  force  P  is  caused  to  act  at 
A  but  in  the  opposite  direction  to  Q  reducing  the  displacement  at  B  to 
zero,  then  Q/P  =  yA/yB* 

In  these  three  relationships  it  is  supposed  that  the  displacements  are 
in  the  directions  of  the  acting  forces. 

Proofs  of  these  relations  and  some  applications  will  be  found  in  a  paper 
by  C.  E.  Larard,  Engineering,  p.  287  (1923). 

§  1-24.  Let  us  next  consider  a  continuous  beam  with  supports  at  A,  B 
and  C.  The  bending  moment  M  at  any  point  in  AB  or  BC  is  the  sum  of 
bending  moment  Ml  of  a  beam  which  is  pin-jointed  at  ABC  and  of  the 
moment  M2  caused  by  the  fixing  moments  at  the  supports.  Let  us  take  B 
as  origin  and  let  12  denote  the  length  BC. 

For  a  beam  which  is  pin-jointed  at  B  and  C  we  have 

M1  =  \w  (I2x  -  x2), 

while  for  a  weightless  beam  with  fixing  moments  MB,  Mc  at  B  and  C 
respectively,  we  have 

M2  =  -  MB  -  (Mc  -  MB)  x/l2. 

Hence       M  =  B2d2y/dx2  =  \w2l2x  -  \w2x2  -  MB  -  (Mc  -  MB)  x/l2. 
Integrating,  we  have 

B2dy/dx  =  ±w2l2x2  -  w2x*/$  -  MBx  -  x2  (Mc  -  MB)/212  -  B2iB, 
where  IB  is  the  value  of  dy/dx  at  x  =  0.   Integrating  again, 

B2y  =  W2l2x3/l2  -  w2x*/24:  -  x2MB/2  -  x9  (Mc  -  MB)/6l2  -  B2iBx. 
When  x  -  12,  y  =  -  y2,  say 

6B2iB  =  ±w2l2*  -  2MB12  -  MC12  +  M^yz. 

*  See  especially  Spofford,  Theory  of  Structure*;  D.  B.  Steinman,  Engineering  Record  (1916); 
G.  E.  Beggs,  International  Engineering,  May  (1922). 


The  Equation  of  Three  Moments  21 

Similarly,  by  considering  the  span  BA,  taking  B  again  as  origin,  but 
in  this  case  taking  x  as  positive  when  measured  to  the  left, 

Eliminating  i&  we  obtain  the  equation 

B^  (MA  +  2MB)  +  BJt  (Mc  +  2MB)  -  J  (^A3S2  +  wJfBJ 

+  6  (B2ll~lyl  +  B^l^y^. 

This  is  the  celebrated  equation  of  three  moments  which  was  given  in 
a  simpler  form  by  Clapeyron*  and  subsequently  extended  for  the  general 
case  by  Heppelt,  WeyrauchJ,  Webb§  and  others ||. 

The  reaction  at  B  is  the  sum  of  the  shears  on  the  two  sides  of  B  and  is 

therefore  7        ,,        ,,  7        ,,        ,, 

%    =  W2*2  +  MB-  MC  ^  wJi      MR  -  MA 

Similarly  for  the  other  supports. 

§  1-25.    When  a  light  beam  or  thin  rod  originally  in  a  vertical  position 
is  acted  upon  by  compressive  forces  P  at  its  ends  (Fig.  4)          | 
the  equation  for  the  bending  moment  is 

M  =  Bd2y/dx2  =  -  Py, 
or  d2y/dx2  +  kzy  =  0, 

where  k2  =  P/B; 

and  if  y  =  0  when  x  —  0  and  when  x  =  a,  the  solution  is 
y  =  A  sin  kx,  where  sin  ka  =  0  or  A  =  0. 

If  ak  <  77,  the  analysis  indicates  that  A  =  0.  A  solution 
with  A  £  0  becomes  possible  when  ak  =  TT.  The  corresponding 
load  P  =  B7T2/a2  is  called  Euler's  critical  load  for  a  rod  pinned 
at  its  ends.  When  P  is  given  there  is  a  corresponding  critical 
lejijgth  a  =  P^/B^Tr. 

To  obtain  these  critical  values  experimentally  great  care 
must  be  taken  to  eliminate  initial  curvature  of  the  rod  and 
bad  centering  of  the  loads.  The  formula  of  Euler  has  been  confirmed  by 
the  experiments  of  Robertson.  In  general  practice,  however,  the  crippling 
load  PC  is  found  to  be  less  than  the  critical  load  P0  given  by  Euler's  formula, 
and  many  formulae  for  struts  have  been  proposed.  For  these  reference 
must  be  made  to  books  on  Elasticity  and  the  Strength  of  Materials. 

In  the  case  of  a  strut  clamped  at  both  ends  there  is  an  unknown  couple 
MQ  acting  at  each  end.  The  equation  is  now 

Bd*y/dx2+  Py=M0, 

*  E.  Clapeyron,  Comptes  Retidus,  t.  XLV,  p.  1076  (1857). 

t  J.  M.  Heppel,  Proc.  lust.  Civil  Engineers,  vol.  xix,  p.  625  (1859-60). 

J  Weyrauch,  Theorie  der  continuierlichen  Trdger,  pp.  8-9. 

§  R.  R.  Webb,  Proc.  Camb.  Phil  Soc.  vol.  vi  (1886).  (Case  Bl  4=  B2.) 

||  M.  Levy,  Statique  graphique,  t.  n  (Paris,  1886).   (Case  yl  4=  0,  yz  4s  0.) 


22  The  Classical  Equations 

and  the  solution  is  of  type 

Py  =  MQ  -f  a  cos  kx  -\-  j$  sin  &#. 

The  boundary  conditions  y  =  0,  dy/rfa  =  0  at  a:  =  a  and  #  =  0  give 
0=0,     a  =  —  MQ,     MQ  sin  &a  =  0,     M0  (1  —  cos  ka)  =  0. 

Hence  either  Jfef0  =  0  or  sin  (to/2)  =  0.  The  critical  load  is  now  given  by 
the  equation  ka  =  2?r  and  is  P0  =  4Brr2/a2. 

When  the  load  P  reaches  the  critical  value  the  rod  begins  to  buckle,  and 
for  a  discussion  of  the  equilibrium  for  a  load  greater  than  PQ  a  theory  of 
curved  rods  is  needed. 

In  the  case  of  a  heavy  horizontal  beam  of  weight  w  per  unit  length  and 
under  the  influence  of  longitudinal  forces  p  at  its  ends,  the  equation 
satisfied  by  the  bending  moment  M  is 


where  k*  =  P/B. 

If  M  =  M0  when  x  =  0  and  M  =  Mj  when  a;  =  a,  the  solution  of  the 
differential  equation  is 


v^  —  M)  sin  ka  =  ( »2 ""  ^o )  sin  ^  (a  ~"  x)  +  ( 7T2  ~  -^i )  s*n  ^ 
Let  us  assume  that  MQ  and  Jf  1  are  both  positive  and  write 

M 0  =  ,  0  sin2  0,     Jkf i  =  ,  0  sin2  ^,     kx  =  a.     k  (a  —  x)  —  B,     ka  =  a  +  B. 
fc2  k*         r 

The  equation  which  determines  the  points  of  zero  bending  moment 
(points  of  inflexion)  is 

sin  (a  -f  0)  =  cos  20  .  sin  j8  -f  cos  2(f>  .  sin  a. 

We  shall  show  that  if  a  and  0  are  both  positive  this  equation  implies  that 
a-f/3>  2  (#  +  </>)  and  so  determines  a  certain  minimum  length  which  must 
not  be  exceeded  if  there  are  to  be  two  real  points  of  inflexion. 

Let  us  regard  6  and  <f>  as  variable  quantities  connected  by  the  last 
equation  and  ask  when  0  +  </>  is  a  maximum.  Writing  z  =  B  -f  <f>  we  have 

-T|  =  1  +  -~f ,  0  =  sin  28  sin  B  -f-  sin  2<A  sin  a  ^ , 

au  au  au 


*z     d*<f>  2sinj8      f 

b"2  =  "^«  ^  ~  ~  -  ^T7T7    c 
O*     dO*          sm  a  sm2  2(f>[_ 


ft         rt/        . 
cos  2   sm  2    -  sin 


When  2$  =  a,  we  have  2<f>  =  j8,  and  these  values  of  6  and  <£  give  a  zero 
value  of  JQ  and  a  negative  value  of  ^  ,  they  therefore  give  us  a  maximum 

value  of  2,  and  so  for  ordinary  values  of  6  and  <f>  we  have  the  inequality 

2  (0  +  <£)  <  a  -f  j8. 


Stability  of  a  Strut  23 

The  position  of  the  points  of  inflexion  is  of  some  practical  interest 
because,  in  the  first  place,  A.  R.  Low*  has  pointed  out  that  instability  is 
determined  by  the  usual  Eulerian  formula  for  a  pin- jointed  strut  of  a  length 
equal  to  the  distance  between  the  points  of  inflexion,  if  these  lie  on  the 
beam,  and  secondly,  if  any  splicing  is  to  be  done,  the  flanges  should  'be 
spliced  at  one  of  the  points  where  the  bending  moment  is  zerof. 

When  P  is  negative  and  so  represents  a  pull  we  may  put  p2  =  -  P/B, 
and  the  solution  is 

(  ~2  ~^~  M }  sinh  pa  =»  (  —  -f  MQ }  sinh  p  (I  —  x)  -f  (    0  4-  M* )  sinh  vx. 
\p*          /  \p£  )  N  \p2          LJ          r 

If  we  write 

2w  ,-       2w 

M0  =  —  sinh-  0,     MI  "-        sinh2  ^6,     px  =  a,     p  (a  -  x)  =  )8,     pa  =  a  +  fi, 
p  p 

a  value  of  x  for  which  M  ==  0  is  determined  by  the  equation 
sinh  (a  -f-  jS)  =  cosh  20  sinh  ^3  -f  cosh  2^>  sinh  a. 
This  equation  implies  that 

a  -f  j3  >  2  (0  -f  (/>). 

For  a  continuous  beam  acted  on  by  longitudinal  forces  at  the  points  of 
support  there  is  an  equation  analogous  to  the  equation  of  three  moments 
which  is  obtained  by  a  method  similar  to  that  used  in  obtaining  the  ordinary 
equation  of  three  moments.  We  give  only  an  outline  of  the  analysis. 

Case  1.       k*=P2/B2,    EG  =  6,     2/3  =  bk,    yA  =  VB  =  yc  =*  0, 


where 


*  .4ero?iaMftcaZ  Journal,  vol.  xvin,  p.  144  (April,  1914).   See  also  J.  Perry,  Phil  Mag.  (March, 
1892);  A.  Morley,  ibid.  (June,  1908);  L.  N.  G.  Filon,  Aeronautics,  p.  282  (Sept.  1919). 
t  H.  Booth,  Aeronautical  Journal,  vol.  xxiv,  p.  563  (1920). 


24  The  Classical  Equations 

There  is  a  corresponding  equation  for  the  bay  BA  which  is  of  length  a, 
if  h2  =  Pi/Bi ,  2a  =  ah,  the  equation  of  three  moments  has  the  form 

aMA  bMc  f,n\  ,   OTIT    !  a  JL  t  \  _.    *  JL  /m'      ^ia3  /  /  \  _L 

-^-/(a)-f  -g-  /(/?)  -f  2^  j^ '0  («)+  ^"^(P)f  =  4^    ^(a)  + 

Case  2.   P  <  0.  The  corresponding  equations  are 

Pj/JJj,     2a  =  ah,     2/?  =  6i, 

O  (a)  +  I  O  (j8)J 


r,  .  .      3  1  -  2a  cosech  2a       A  ,  .       3  2a  coth  2a  —  1 
*•<«)  =  2-~  --,  -  -,     OW-i  -  ^i—  • 

V  (a)  = 

The  functions/  (a),  P  (a),  etc.,  have  been  tabulated  by  Berry*  who  has 
also  given  a  complete  exposition  of  the  analysis.  These  equations  are  much 
used  in  the  design  of  airplanes  built  of  wood. 

EXAMPLES 

1.  Find  the  crippling  load  for  a  rod  which  is  clampod  at  one  end  and  pinned  at  the  other. 

2.  Prove  that  jn  the  case  of  a  uniform  light  beam  of  length  a  with  a  concentrated  load 
W  at  x  =  |  the  solution  can  be  written  in  the  form 

«      2Wa  I  .    nx  .    TT£      1    .    2nx  .    2n{         \ 
M  =        -  1  sin  —  sin  ---  h  ^  sin  —  sin        +...), 
7T2    \       a         a      22         a  a  ) 

TTX  .    TT£       1 

a~  8m  a  +  2* 

when  the  beam  is  pinned  at  both  ends.  The  corresponding  formulae  for  a  uniformly  dis- 
tributed load  are 

,    .        4tW  f  .     TTX        1     .     3irX        1     .     5nX  \ 

w(x)  =     -  (sin—  -f  Osm  ----  1-  _  sin         +...), 
TT  \       a       3         a       5         a  ) 

,. 

M  = 


va2  (  .    nx       1    .    3nx      1    .    57ra;          \ 
-„-      sm  ----  h  jr-  sin   ---  h  ^>  sin  ---  h  ...  )  , 

T3  \       a       33         a        53          a  / 

4wa*  f  .     TTX        1     .     STTX        1     .     STTX  \ 

y  =  D  .     sin  ---  h  5-  sm  —  -f-  K5  sin  ---  -f  ...    . 

Bn5  \       a       3°         a        55         a  / 


[Timoshenko  and  Lessells  Applied  Elasticity,  p.  230.] 

3.  Find  the  form  of  a  strut  pinned  at  its  ends  and  eccentrically  loaded  at  its  ends  with 
compressional  loads  P. 

4.  The  Green's  function  for  the  differential  expressionf 


*  Trans.  Roy.  Aeronautical  Soc.  (1919).  The  tables  are  given  also  inPippard  and  Pritchard*a 
Aeroplane  Structures,  App.  I  (1919). 

f  Examples  4-6  are  taken  from  a  paper  by  A.  Myller,  2$as.  Oottingen  (1906). 


End'  Conditions  25 

and  the  end  conditions  u  (0)  -  u  (I)  =  u'  (0)  -  u'  (1)  -  0  is 


~  fl\*-\  &>•        *M- 

^  «_  [l*jk 

(z)>        P-  Jo  £<*")' 


where 


5.    If  in  the  last  example  the  boundary  conditions  are  u  (0)  =  u'  (0)  =  u"  (  1  )  =  u"'  (  1  )  =  0 
the  Green's  function  is 

*  '  *  ~  4 


6.   When    the    end    conditions    are    u  (0)  =  w"  (0)  =  u  (1)  =.  u"  (1)  =  0,    the    Green's 
function  is  0  (#,  (  ),  where 


-  (a  -  4)3  +  4y)*f  -  (0  -  2y)(x  +  f  )  -  y. 


§  1-31.  jFVee  undamped  vibrations.  Whenever  a  particle  performs  free 
oscillations  in  a  straight  line  under  the  influence  of  a  restoring  force  pro- 
portional to  the  distance  from  a  fixed  point  on  the  line  the  equation  of 

motion  is 


mx  =  - 


where  m  is  the  mass  of  the  particle  and  X\L  is  the  restoring  force.  Writing 
M  =  4«m,  we  have  £ 


an  equation  which  has  already  been  briefly  considered.   The  general 
solution  is 


gn 


where  ^4  and  JS  are  arbitrary  constants.    Writing  k  =  27m,  A  =  a  sin 
B  =  a  cos  0  we  have 

x  =  a  sin  (27m£  -f  0). 


The  quantity  a  specifies  the  amplitude,  n  the  frequency  and  27fnt  -f  0 
the  phase  of  the  oscillation.  The  angle  0  gives  the  phase  at  time  t  =  0. 
The  period  of  vibration  T  may  be  found  from  the  equations 

kT  -  277,     nT  =  1. 

This  type  of  vibration  is  called  simple-harmonic  vibration  because  it  is 
of  fundamental  importance  in  the  theory  of  sound.  The  vibrations  of  solid 
bodies  which  are  almost  perfectly  rigid  are  often  of  this  type,  thus  the 
end  of  a  prong  of  a  tuning  fork  which  has  been  properly  excited  moves  in 
a  manner  which  may  be  described  approximately  by  an  equation  of  this 


26  The  Classical  Equations 

type.  The  harmonic  vibrations  of  the  tuning  fork  produce  corresponding 
vibrations  in  the  surrounding  air  which  are  of  audible  frequency  if 

24  <  n  <  24000. 
The  range  of  frequencies  used  in  music  is  generally 

40  <  n  <  4000. 

The  differential  equation  (I)  may  be  replaced  by  two  simultaneous 
equations  of  the  first  order 

x  +  ky=0,     y-kx=0  (II) 

which  imply  that  the  point  Q  with  rectangular  co-ordinates  (x,  y)  moves 
in  a  circle  with  uniform  speed  ka.   We  have,  in  fact,  the  equation 

xx  4-  yy  =  0, 

which  signifies  that  x2  -f-  y2  is  a  constant  which  may  be  denoted  by  a2. 
There  is  also  an  equation 

#2  -f  I/*  =   k*  (3.2  _j_   y2)   =   £2a2? 

which  indicates  that  the  velocity  has  the  constant  magnitude  ka.  The 
solution  of  the  simultaneous  equations  may  be  expressed  in  the  form 

x  =  a  cos  a,     y  =  a  sin  a, 
where  a  —  Zrrnt  4-  6  —  ~ ;  • 

Simultaneous  equations  of  type  (II)  describe  the  motion  of  a  particle 
which  is  under  the  influence  of  a  deflecting  force  perpendicular  to  the 
direction  of  motion  and  proportional  to  the  velocity  of  the  particle.  The 
equations  of  motion  are  really 

x  -f-  ky  —  0,     y  —  kx  =  0, 

but  an  integration  with  respect  to  t  and  a  suitable  choice  of  the  origin  of 
co-ordinates  reduces  them  to  the  form  (II).  The  equations  may  also  be 

written  in  the  form  7  , 

u  +  kv  =  0,     v  —  leu  =  0, 

where  (u,  v)  are  the  component  velocities. 

If  the  deflecting  force  mentioned  above  is  the  deflecting  force  of  the 
earth's  rotation  the  deflection  is  to  the  right  of  a  horizontal  path  in  the 
northern  hemisphere  and  to  the  left  in  the  southern  hemisphere.  If  the 
angle  <f>  represents  the  latitude  of  the  place  and  o>  the  angular  velocity  of  the 
earth's  rotation,  the  quantity  k  is  given  by  the  formula 

k  —  2a>  sin  (f>. 

When  the  resistance  of  the  air  can  be  neglected,  the  suspended  mass  M 
of  a  pendulum  performs  simple  harmonic  oscillations  after  it  has  been 
slightly  displaced  from  its  position  of  equilibrium.  The  vertical  motion  is 
now  so  small  that  it  may  be  neglected  and  the  acceleration  may,  to  a  first 


Simple  Periodic  Motion  27 

approximation,  be  regarded  as  horizontal  and  proportional  to  the  horizontal 
component  of  the  pull  P  of  the  string.  We  thus  have  the  equation  of  motion 

Mix  =  -  Px  =  -  Mgx, 

where  /  is  the  length  of  the  string  and  g  the  acceleration  of 
gravity.  The  mass  of  the  string  is  here  neglected.  With  this  Q 
simplifying  assumption  the  j  endulum  is  called  a  simple  pendu- 
lum. In  dealing  with  connected  systems  of  simple  pendulums 
it  is  convenient  to  use  the  notation  (I,  M)  for  a  simple  pendu- 
lum whose  string  is  of  length  I  and  whose  bob  is  of  mass  M 
(Fig.  5). 

If  the  string  and  suspended  mass  are  replaced  by  a  rigid 
body  free  to  swing  about  a  horizontal  axis  through  the  point 
0,  the  equation  of  motion  is  approximately 

19  =  -  MgliO, 

where  7  is  the  moment  of  inertia  of  the  body  about  the  horizontal  axis 
through  0  and  h  is  the  depth  of  the  centre  of  mass  below  the  axis  in  the 
equilibrium  position  in  which  the  centre  of  mass  is  in  the  vertical  plane 
through  0.  Writing  Mhg  =  Ik2  the  equation  of  motion  becomes 

0  +  ]*0  -  0, 

and  the  period  of  vibration  is  27r/k,  a  quantity  which  is  independent  of 
the  angle  through  which  the  pendulum  oscillates. 

This  law  was  confirmed  experimentally  by  Galileo,  who  showed  that 
the  times  of  vibration  of  different  pendulums  were  proportional  to  the 
square  roots  of  their  lengths.  The  isochronism  of  the  pendulum  for  small 
oscillations  was  also  discovered  by  him  but  had  been  observed  previously 
by  others.  When  the  pendulum  swings  through  an  angle  which  is  not 
exceedingly  small  it  is  better  to  use  the  more  accurate  equation 

0  +  £2  sin  0=0, 

which  may  be  derived  by  resolving  along  the  tangent  to  the  path  of  the 
centre  of  gravity  G  or  by  differentiating  the  energy  equation 

=  Mgh  (cos  0  -  cos  a), 


which  is  written  down  on  the  supposition  that  the  velocity  of  0  is  zero 
when  0  =  a.   With  the  aid  of  the  substitution 

sin  (|0)  =  sin  (Ja)  sin</>, 
this  equation  may  be  written  in  the  form 

^2^  p  [i  _  sin2  \a  sin2  <£]. 

As  0  varies  from  —  a  to  a,  <f>  varies  from  —  «  ^°  5  >  an(^  so 


28  The  Classical  Equations 

swing  from  one  extreme  position  (0  =  —  a)  to  the  next  extreme  position 
(0  =  «)  is 

2    (1  -  sin2  \a  sin2 </»)"*  d*j>. 

When  a  is  small  the  period  T  is  given  approximately  by  the  formula 
kT  =  27r(l  -f  Jsin2  J«) 

and  depends  on  a,  so  that  there  is  not  perfect  isochronism. 

This  fact  was  recognised  by  Huygens  who  discovered  that  perfect 
isochronism  could  theoretically  be  secured  by  guiding  the  string  (or  other 
flexible  suspension)  with  the  aid  of  a  pair  of  cycloidal  cheeks  so  as  to  make 
the  centre  of  gravity  describe  a  cycloidal  instead  of  a  circular  arc.  This 
device  has  not,  however,  proved  successful  in  practice  as  it  introduces 
errors  larger  than  those  which  it  is  supposed  to  remove*.  More  practicable 
methods  of  securing  isochronism  with  a  pendulum  have  been  described  by 
Phillipsf. 

§  1-32.    Simultaneous  equations  of  type 

Lx  -j-  My  -f  Lm2x  =  0, 
MX  +  Nij  -f  Nn*y  =  0, 

in  which  L,  M,  Ny  my  n  are  constants,  occur  in  many  mechanical  and 
electrical  problems.  When  the  coefficient  M  is  zero  the  co-ordinates  x  and 
y  oscillate  in  value  independently  with  periods  Sir/m  and  27r/n  respectively, 
but  when  M  =£  0  the  assumption 

x  =  p2MAelpt,     y  =  LA  (m2  -  p2)  eipt 
gives  the  equation 

p*  (1  -  y2)  -  p2  (m2  4-  n2)  +  m2n2  =  0, 
where  y2  =  M2/LN. 

This  quantity  y  is  called  the  coefficient  of  coupling! . 
When  m  =5^  ?i  we  have  -         „   . 

V2D4 
7)2_m2==        Yf 

*  p2-n2' 

and  when  y  is  small  the  value  of  p  which  is  close  to  m  is  given  approximately 

by  the  equation 

>p2  _  m2  =  -JT™      _  p  2  say. 
*  m2-  n2         r  >     j 

A  simple  harmonic  oscillation  of  the  #-co-ordinate,  with  a  period  close 
to  the  free  period  2-jT/m,  is  accompanied  by  a  similar  oscillation  of  the 

*  See  R.  A.  Sampson^  article  on  "  Clocks  and  Time-Keeping"  inDictionary  of  Applied  Physics, 
vol.  m. 

t  Comptes  Rendus,  t.  oxn,  p.  177  (1891). 

J  See,  for  instance,  E.  H.  Barton  and  H.  Mary  Browning,  Phil.  Mag.  (6),  vol.  xxxiv,  p.  246 
(1917). 


Simultaneous  Linear  Equations  29 

y-co-ordinate  with  the  same  period  but  opposite  phase.  The  amplitude  of 
the  ^-oscillation  is  proportional  to  y2.  Now  let  p1  be  the  greater  of  the  two 
values  of  p.  If  m  >  n  we  have  pl  >  m  but  if  m  <  n  we  have  p2  <  m.  The 
effect  of  the  coupling  is  thus  to  lower  the  frequency  of  the  gravest  mode  of 
vibration  and  to  raise  the  frequency  of  the  other  mode  of  simple  harmonic 
vibration.  If  m  =  n  the  equation  for  p2  gives 

p2  =  m2  ±  yp2, 

and  the  effect  of  the  coupling  is  to  make  the  periods  of  the  two  modes 
unequal.  In  the  general  case  we  can  say  that  the  effect  of  the  coupling 
is  to  increase  the  difference  between  the  periods.  The  periods  may,  in  fact, 
be  represented  geometrically  by  the  following  construction : 

Let  OA,  OB  represent  the  squares  of  the  free  periods,  the  points  0,A,B 
being  on  a  straight  line.  Now  draw  a  circle  F  on  AB  as  diameter  and  let 
a  larger  concentric  circle  cut  the  line  OA  B  in  U  and  V  \  the  distances  OC7, 
O  V  then  represent  the  squares  of  the  periods  when  there  is  coupling.  If  a 
tangent  from  0  to  the  circle  F  touches  this  circle  at  T  and  meets  the  larger 
circle  in  the  points  M  and  L  the  coefficient  of  coupling  is  represented  by 
the  ratio  TL/TO  (Fig.  6). 

So  long  as  0  lies  outside  the  larger  circle  it  is  evident  that  the  difference 
between  the  periods  is  increased  by  the  coupling,  but  when  y  >  1  the  point 
0  lies  within  the  larger  circle  and  the  difference  between  the  periods  de- 
creases to  zero  as  the  radius  CU  of  this  circle  increases  without  limit.  There 
is  thus  some  particular  value  of  the  coupling  for  which  the  difference 
between  the  periods  has  the  original  value,  both  periods  being  greater  than 
before. 

When  y  =  1  the  equations  of  motion  may  be  written  in  the  forms 

Lx  +  My  +  Lm2x  =  0,     Lx  +  My  +  Mn2y  =  0, 

and  imply  that  Lm2x  =  Mn2y.  There  is 
now  only  one  period  of  vibration.  The 
cases  y>  \  are  not  of  much  physical 
interest  as  the  values  of  the  constants 
are  generally  such  that  M 2  <  LN,  this 
being  the  condition  that  the  kinetic 
energy  may  be  always  positive. 

Equations  of  the  present  type  occur 
in  electric  circuit  theory  when  resist- 
ances are  neglected.  In  the  case  of  a 
simple  circuit  of  self-induction  L  and  lg* 

capacity  C  furnished,  say,  by  a  Ley  den  jar  in  the  circuit,  the  charge  Q 
on  the  inside  of  the  jar  fluctuates  in  accordance  with  the  equation 

®  =  0 


30  The  Classical  Equations 

when  the  discharge  is  taking  place.  The  period  of  the  oscillations  is  thus 

T  =  2n  W(LC). 

This  is  the  result  obtained  by  Lord  Kelvin  in  1857  and  confirmed  by 
the  experiments  of  Fedderson  in  1857.  The  oscillatory  character  of  the 
discharge  had  been  suspected  by  Joseph  Henry  from  observations  on  the 
magnetization  of  needles  placed  inside  a  coil  in  a  discharging  circuit. 

In  the  case  of  two  coupled  circuits  (Ll9  C^),  (L2,  (72)  the  mutual  induction 
M  needs  to  be  taken  into  consideration  and  the  equations  for  free  oscil- 
lations are  ~ 

L&  +  MQ,  +    /  -  0, 


§  1-33.  The  Lagrangian  equations  of  motion.  Consider  a  mechanical 
system  consisting  of  /  material  points  of  which  a  representative  one  has 
mass  m  and  co-ordinates  x,  y,  z  at  time  /.  Using  square  brackets  to  denote 
a  summation  over  these  material  points,  we  may  express  d'Alembert's 
principle  in  the  Lagrangian  form 

[m  (xSx  +  y8y  -f  5  82)]  -  [XSx  +  Y8y  +  ZSz], 

where  8x9  §//,  8z  are  arbitrary  increments  of  the  co-ordinates  which  are 
compatible  with  the  geometrical  conditions  limiting  the  freedom  of  motion 
of  the  system.  On  account  of  these  conditions,  the  number  of  degrees  of 
freedom  is  a  number  N9  which  is  less  than  3J,  and  it  is  advantageous  to 
introduce  a  set  of  "generalised11  co-ordinates  ql9  g2,  ...  gv  which  are  inde- 
pendent in  the  sense  that  any  infinitesimal  variation  8qs  of  qs  is  compatible 
with  the  geometrical  conditions.  These  conditions  may,  indeed,  be  expressed 
in  the  form 

#•=/(?!  >?2»   •••  <LV>  0>       y  =  9  (?1>?2>   •••  1\'>0>       *  =   M<?1,?2>  •'•  <7A>0- 

Using  the  sign  S  to  denote  a  summation  from  1  to  N,  a  prime,  to  denote 
a  partial  differentiation  with  respect  to  t  and  a  suffix  s  to  denote  a  partial 
differentiation  of  x9  y  or  z  with  respect  to  qs,  we  have  the  equations 

x  =  x'  -f  Zxsq89     Sx  =  I<x38qS9 


where  the  quantities  Q(s)  may  be  called  generalised  force  components 
associated  with  the  co-ordinates  q.  The  first  of  these  equations  shows  that 
xs  is  also  the  partial  derivative  of  x  with  respect  to  qs  and  so  if  the  kinetic 
energy  of  the  system  is  T9  where 

2T  =  [m  (x2  4-  y2  +  z2)], 
we  shall  have  -_ 

Cl  r 


Lagrange's  Equations          *  31 

where  p8  is  the  generalised  component  of  momentum.  Since 
dx      Sxf      v        .       dxr 
Sq^Wr^  "Xr*qs^~dt  =s*r' 

and  -.--  (xxs)  =  xxs  -f  #xg, 

rt* 

we  have 

[m  (xSx  +  ySy  +  282)]  -  S8gs     ,  m  (^  +  yys  +  22,)  -  m  (xxs  -f  yy,  +  zzs) 

iai  .  J 


and  so  Lagrange's  principle  may  be  written  in  the  form 

dT\       v-/ir<,^ 

-**  *9" 


On  account  of  the  arbitrariness  of  the  increments  8qs  this  relation  gives 
the  Lagrangian  equations  of  motion 

dfST^dT^ 
dt\dqs)      dqs      ^     ' 

If  there  is  a  potential  energy  function  F,  which  can  be  expressed  simply* 
in  terms  of  the  generalised  co-ordinates  </,  we  may  write 


and  the  equations  of  motion  take  the  simple  form 

d  i  dL\      dL 


The  quantity  L  is  called  the  Lagrangian  function. 
Introducing  the  reciprocal  function 

T-x(adT]      T 

7-M?*a$J      ' 

,  dT      _  f  .   d  /ST\      ..  dT)      „  /..  ST      .  dT\ 

we  have  --  =  S        ,  L          <   ,-.-   -  S  U        +       - 


Hence  we  have  the  energy  equation 

T7  -f  F  =  constant. 

When  the  functions  /,  g  and  h  do  not  contain  the  time  explicitly  we 
have  on  account  of  Euler's  theorem  for  homogeneous  functions 


The  Lagrangian  equations  of  motion  may  be  replaced  by  another  set  of 
equations  for  the  quantities  p  and  q.  For  this  purpose  we  introduce  the 
Hamiltonian  function  H  defined  by 

=  -  L+  ^p8qB- 


32  The  Classical  Equations 

If  we  always  consider  H  as  a  function  of  the  quantities  q8  and  p8  but 
L  as  a  function  of  q8  and  qa,  we  have 


Thus  *H-=q.,    m=-^~, 

consequently  the  equations  of  motion  can  be  expressed  in  the  Hamiltonian 


^_ 

dt  ~~  dp, '       dt   ~       dq, * 

Systems  of  equations  whose  solutions  represent  superposed  simple 
harmonic  vibrations  are  derived  from  the  Lagrangian  equations  of  motion 
of  a  dynamical  system 

d  /dT\  _  dT  __  _  SV 
dt  \dqj     dqs  ~      dqs ' 

5=  1,2,  ...  AT, 

whenever  the  kinetic  energy  T,  and  the  potential  energy  V,  can  be  ex- 
pressed for  small  displacements  and  velocities  in  the  forms 

N       N 

2T=    S     2  amnqmqn, 

m-l n-1 

N      N 

2V  =   S     S  cmn?m?n 

7/1-1   71—1 

respectively,  where  the  constant  coefficients  amn  and  cmn  are  such  that  T 
and  F  are  positive  whenever  the  quantities  qm,  qm  do  not  all  vanish.  For 
such  a  system  the  equations  (A)  give  the  differential  equations 

N 

£     Knrtfn  +  C«in?»)  ^  °> 
n-1 

m-  1,  2,  ...  N. 

Multiplying  by  um,  where  um  is  a  constant  to  be  determined,  and 
summing  with  respect  to  m,  the  resulting  equation  is  of  type 

v  +  k*v=  0,  (B) 

if  the  quantities  um  are  such  that  for  each  value  of  n 

N  N 

S  umcmn  -  fc2  S  wroaww  -  P6n,  say. 

?n«l  m-l 

The  corresponding  value  of  v  is  then 

N 

v=   S  6n?n. 

n-l 


Normal  Vibrations  33 

Now  when  the  quantities  um  are  eliminated  from  the  linear  homo- 
geneous equations  N 

2  (cmn  -  k2amn)  um  =  0,  (C) 

?n  =  l 

we  obtain  an  algebraic  equation  of  the  A7th  degree  for  k2.  With  the  usual 
method  of  elimination  this  equation  is  expressed  by  the  vanishing  of  a 
determinant  and  may  be  written  in  the  abbreviated  form 

I  c       —  k2n       I  —  0 
I  ^mn         K  amn  \   —  u» 

To  show  that  the  values  of  k2  given  by  this  equation  are  all  real  and 
positive,  we  substitute  k2  =  h  -f  ij,  um  =  vm  -f  iwm  in  equation  (C). 
Equating  the  real  and  imaginary  parts,  we  have 

N  N 

2  (cmn  -  Juimn)  vm  +  j  2  amnwm  =  0, 

m  —  1  m  —  1 

AT  iv 

2  (cmn  -  Aamn)  wn-j   2  amnt?w  -  0. 

ra  =  1  ?7i  —  1 

Multiplying  these  equations  by  wn  and  —  vn  respectively,  adding  and 
summing,  we  find  that 

j   2  amn  (wmwn  -f  vmvn)  =  0. 

in,  n 

The  factor  multiplying  j  is  a  positive  quadratic  form  which  vanishes 
only  when  the  quantities  vn,  wn  are  all  zero,  hence  we  must  have  j  —  0  and 
this  means  that  k2  is  necessarily  real.  That  k  is  necessarily  positive  is  seen 
immediately  from  the  equation 

2  cmnumun  =  k2  2  amnumun, 

m,  n  m,n 

which  involves  two  positive  quadratic  forms. 

If  um  (A^),  um  (k2)  are  values  of  um  corresponding  to  two  different  values 
of  k  we  have  the  equations 

2    (Cmn  -  *!«««»)  Um  (*i)  =   0, 

7/J-l 

N 

S  (cwn  -  k22amn)  um  (k,)  =  0. 

TO-l 

Multiplying  these  by  wn  (A:2),  wn  (^2)  respectively  and  subtracting  we 
find  that  (V  _  V)  s  a^Wm  (Aj)  ^  (kj  =  Q (D) 

m,n 

Denoting  the  constant  6W  associated  with  the  value  k  by  6n  (^),  we  see 
from  the  last  equation  that  if  k2  =£  kl9 

S    6n  (*i)  ^n  (*i)  =   0. 
n-1 

On  the  other  hand, 

N 

2  6n  fa)  un  (k^  =   2  amnum  (*J  wn  (^), 

n»  1  m,  n 


34  The  Classical  Equations 

and  is  an  essentially  positive  quantity  which  may  be  taken  without  loss  of 
generality  to  be  unity  since  the  quantities  un  (k^  contain  undetermined 
constant  factors  as  far  as  the  foregoing  analysis  is  concerned. 

Using  the  symbol  v  (k,  t)  to  denote  the  function  v  corresponding  to  a 
definite  value  of  k,  we  observe  that  if 

N 

qm=    S   v(k8,t)  Ams, 

8-1 

N 

we  must  have  S  bn  (k3)  \ms  =  0         n±m 

s-l 

=  1         n  =  ra. 
Multiplying  by  un  (kr)  and  summing  with  respect  to  n  we  find  that 

*mr  =  Um  (kr). 

N 

Hence  qm=   I,  um  (ks)  v  (ks,  t). 

.9-1 

This  expresses  the  solution  of  our  system  of  differential  equations  in 
terms  of  the  simple  harmonic  vibrations  determined  by  the  equations  of 
type  (B).  The  analysis  has  been  given  for  the  simple  case  in  which  the 
roots  of  the  equation  for  k2  are  all  different  but  extensions  of  the  analysis 
have  been  given  for  the  case  of  multiple  roots. 

The  relation  (D)  may  be  regarded  as  an  orthogonal  relation  in  general- 
ised co-ordinates.  When 

amn=  0,     m±n,     amm=  1, 

the  relation  takes  the  simpler  form 
.v 

2  um  (kv)  um  (kq)  =  0         p  +  q. 

m  -1 

§  1-34.  An  interesting  mechanical  device  for  combining  automatically 
any  number  of  simple  harmonic  vibrations  has  been  studied  by  A.  Gar- 
basso*. 

A  small  table  of  mass  m  is  supported  by  four  light  strings  of  equal 
length  /  so  that  it  remains  horizontal  as  it  swings  like  a  pendulum.  The 
table  is  attached  at  various  points  to  n  simple  pendulums  (ls,  ras), 
s  =  1,  2,  ...  n.  Each  string  is  regarded  as  light  and  is  supposed  to  oscillate 
in  a  vertical  plane  and  remain  straight  as  the  apparatus  oscillates. 

Specifying  the  configuration  of  the  apparatus  by  the  angular  variables 
$o>  ^i  >  •••  ®n  we  have  in  a  small  oscillation 


V  = 

*  Vorlesungen  uber  Speklroskopie,  p.  65;  Torino  Atti,  vol.  XLIV,  p.  223  (1908-9).  The  case  in 
which  n  =  2  has  been  studied  in  connection  with  acoustics  by  Barton  and  Browning,  Phil.  Mag. 
(6),  vol.  xxxiv,  p.  246  (1917);  vol.  xxxv,  p.  62  (1918);  vol.  xxxvi,  p.  36  (1918)  and  by  C.  H.  Lees, 
ibid.  vol.  XLVHI  (1924). 


Compound  Pendulum 
The  equations  of  motion  are 


ms 


8=1 


n       \ 

2  ms  }  00  =  0, 

-1  / 


35 


(8=  1,2,  ...n). 

n 

Writing  ma  =  csm0,  S  cs  =  c,  the  equation  for  k2  is  in  this  case 


+  C)  - 
-/0P 

-  I0kz 


-  gc,          -  gc2    ...  -  gcn       |  =o, 
-Z^  o         ...         0          | 

0          g-l2k*...        0         I 


0 


or 


g- 


0 


Expanding  the  determinant  we  obtain  the  equation 


where 


/  (V)  =  II  (g  -  l,k*). 

.s-1 


Now        (10  -  ls)  [g  -  (10  +  ls)  k*}  =  la(g-  I0lc*)  -l,(g-  l,k*), 
consequently  the  equation  for  k2  can  be  written  in  the  form 

/(*»)  \(g  -  kk*)  fl  +  gk  2  j-  -  r~f—  -rJ  -  g  S     C8's  1  =  0. 

(  L  S-l   "O  ~    18)   \(J  —    LS^    )J  8-1  ^0  ~   ^S) 

If  the  mass  of  each  pendulum  is  so  small  in  comparison  with  that  of  the 
table  that  we  may  neglect  terms  of  the  second  order  in  the  quantities  cs, 
the  equation  may  be  written  in  the  form 

(g  -  U«  -  17  S    ±l>  }  0    (g  -  l,k*  +  ^-)  . 

\  .s-1  ^0  ~  1J  .s-1     \  ^0  ~~  's/ 

Hence  the  periods  of  the  normal  vibrations  are  approximately 


-  1,2,  ...n). 


3-2 


36  The  Classical  Equations 

If  IQ  >  18  the  period  of  the  5th  pendulum  is  decreased  by  attaching  it  to 
the  table.   If  10  <  ls  the  period  is  increased. 

EXAMPLES 

1.  A  simple  pendulum  (6,  N)  is  suspended  from  the  bob  of  another  simple  pendulum 
(a,  M  )  whose  string  is  attached  to  a  fixed  point.   Prove  that  the  equations  of  motion  for 

small  oscillations  are  ^Tv    „..       ^T     •• 

(M  +  N)  a2d  +  Nab<f>  +  (M  +  N)  gaS  =  0, 

Nab'B  +  Nb*j>  +  Ngb<f>  =  0, 
where  B  and  <£  are  the  angles  which  the  strings  make  with  the  vertical. 

2.  Prove  that  the  coefficient  of  coupling  of  the  compound  pendulum  in  the  last  example 
is  given  by 


_    ___ 

M  +  N' 

3.  Prove  that  it  is  not  possible  for  the  centre  of  gravity  of  the  two  bobs  to  remain  fixed 
in  a  simple  type  of  oscillation. 

4.  A  simple  pendulum  (/,  'M)  is  suspended  from  the  bob  of  a  lath  pendulum  which  is 
treated  as  a  rigid  body  with  a  moment  of  inertia  different  from  that  of  the  bob.   Find  the 
equations  of  motion  and  the  coefficient  of  coupling. 

5.  A  simple  pendulum  (I,  M)  is  attached  to  a  point  P  of  an  elastic  lath  pendulum  which 
is  clamped  at  its  lower  end  and  carries  a  bob  of  mass  N  at  its  upper  end.    At  time  t  the 
horizontal  displacements  of  M,  N  and  P  are  y,  z  and  az  respectively,  a  being  regarded  as 
constant.   By  adopting  the  simplifying  assumption  that  a  horizontal  component  force  F  at 
P  gives  N  the  same  horizontal  acceleration  as  a  force  aF  acting  directly  on  N,  obtain  the 
equations  of  motion 

IMy  +  Mg  (y  -  az)  -  0,     INz  +  lNn*z  =  Mga  (y  -  az), 
and  show  that  the  coefficient  of  coupling  is  given  by  the  equation 

a? 


~  Nri*  +  Mm**9 
where  lm2  =  g. 

[L.  C.  Jackson/PAtf.  Mag.  (6),  vol.  xxxix,  p.  294  (1920).] 

ft.  Two  masses  m  and  m'  are  attached  to  friction  wheels  which  roil  on  two  parallel 
horizontal  steel  bars.  A  third  mass  J/,  which  is  also  attached  to  friction  wheels  which  roll 
on  a  bar  midway  between  the  other  two,  is  constrained  to  lie  midway  between  the  other  two 
masses  by  a  light  rigid  bar  which  passes  through  holes  in  swivels  fixed  on  the  upper  part  of 
the  masses.  The  masses  m  and  m'  are  attached  to  springs  which  introduce  restoring  forces 
proportional  to  the  displacements  from  certain  equilibrium  positions.  Find  the  equations  of 
motion  and  the  coefficient  of  coupling. 

This  mechanical  device  has  been  used  to  illustrate  mechanically  the  properties  of  coupled 
electric  circuits.  [See  Sir  J.  J.  Thomson,  Electricity  and  Magnetism,  3rd  ed.  p.  392  (1904); 
W.  S.  Franklin,  Electrician,  p.  556  (1916).] 

7.  Two  simple  pendulums  (/lf  Jfj),  (/2,  M2)  hang  from  a  carriage  of  mass  M  which,  with 
the  aid  of  wheels,  can  move  freely  along  a  horizontal  bar.  Prove  that  the  equations  of  motion 

are  (M  +  M1  +  M2)x  +  MJ&  +  M212§2  =  0, 


Quadratic  Forms  37 

Hence  show  that  the  quantities  B19  92  can  be  regarded  as  analogous  to  electric  potential 
differences  at  condensers  of  capacities  M^g  and  M2g,  the  quantities  Mlllgdl  and  Mzl2g6z  as 
analogous  to  electric  currents  in  circuits,  the  quantities 

, 
) 

as  analogous  to  coefficients  of  self-induction  and  [(Ml  +  M2  +  M )  g2]~*  as  analogous  to  a 
coefficient  of  mutual  induction. 

[T.  R.  Lyle,  Phil.  Mag.  (6),  vol.  xxv,  p.  567  (1913).] 

8.  A  simple  pendulum  of  length  Z,  when  hanging  vertically,  bisects  the  horizontal  line 
joining  the  knife  edges.  When  the  pendulum  oscillates  it  swings  freely  until  the  string  comes 
into  contact  with  one  of  the  knife  edges  and  then  the  bob  swings  as  if  it  were  suspended  by 
a  string  of  length  h.  Assuming  that  the  motion  is  small  and  that  in  a  typical  quarter  swing 

10  +  go  =  0  for  0  <  t<  T, 

hd  +  00  =  0  for  r  <  t  <  T, 
prove  that  the  quarter  period  T  is  given  by  the  equation 

m  cot  n  (T  —  r)  =  n  tan  mr, 
where  g  =  Im?  =  hri2. 

§  1-35.    Some  properties  of  non-negative  quadratic  forms*.   Let 

nt  n 

g  =    2  grsxrxs 
i,  i 

be  a  quadratic  form  of  the  real  variables  xl9  ...  xn,  which  is  negative  for  no 
set  of  values  of  these  variables,  then  there  are  n  linear  forms 


1 
with  real  coefficients  prg  such  that 

n      /    n 

g  =   S    ^  2  prsx, 
This  identity  gives  the  relation 

n 

glk  =    S  ^rt^rfc 
i 

which  can  be  regarded  as  a  parametric  representation  of  the  coefficients  in 
a  non-negative  form. 

This  result  may  be  obtained  by  first  noting  that  gss  is  not  negative,  for 
g^  is  the  value  of  g  when  xr  =  0,  r  £  s  and  xs  =  1 .  If  the  coefficients  gr9 
are  not  all  zero  the  coefficients  gss  are  not  all  zero,  because  if  they  were  and 
if,  say,  012  <0  a  negative  value  of  g  could  be  obtained  by  choosing  x1  =  1, 
x2  =  T  1,  #3  =  ...  o:n  =  0. 

We  may,  then,  without  loss  of  generality  assume  that  there  is  at  least 
one  coefficient  gu  of  the  set  g88  which  is  positive. 

*  L.  Fejer,  Math.  Zeits.  Bd.  i,  S.  70  (1918). 


38  The  Classical  Equations 

Writing  pnp18  =  gls      s  =  1,  2,  ...  n, 

n 
2j  =  S  plsXs, 

8-1 

<7<"  =  g  -  «,», 

it  is  easily  seen  that  the  quadratic  form  g(l}  does  not  depend  on  xv  g(l}  is 
moreover  non-negative  because  if  it  were  negative  for  any  set  of  values 
of  #2,  #3,  ...  xn  we  could  obtain  a  negative  value  of  g  by  choosing  xl  so  that 
2l  =  0.  ' 

Since  g(l)  is  non-negative  it  either  vanishes  identically  or  the  coefficient 
of  at  least  one  of  the  quantities  #22,  x32,  ...  xn2  in  f/{1)  must  be  positive. 
Let  us  suppose  that  gr22(1)  is  positive  and  write 

(1>  r  =    2,   3,...tt/ 


0(2)  _  yd)   _  z22. 

Continuing  this  process  it  is  found  that  g  =  2^+  z224-  .  .  .zn2,  where  the 
linear  forms  ^  ,  z2,  .  .  .  zn  are  not  all  zero  ;  it  is  also  found  that  none  of  the 
quantities  gn,  <722(1),  fe*2*,  ...  are  negative  and  that  all  these  quantities, 
except  the  first,  are  ratios  of  leading  diagonal  minors  in  the  determinant 

|   9rs   |  , 

and  are  not  all  zero. 

n,  n 

Now  let  h  =   £  htkxtxk 

i,  i 

be  a  second  non-negative  form,  and  let 

n 
hik  =    S  Jat^fc 

be  its  parametric  representation,  then* 

n,  n  n,n        n 

2  gtkhtk=  2  (  S 

1,1  1,1     \<r-l 

If  0>  2/i,  ^2»  •••  2/n  are  arbitrary  real  quantities, 

2  (x,  -  6ysy 

is  never  negative.  Regarding  this  as  a  quadratic  expression  in  0  it  is 
readily  seen  that  the  quadratic  form 

A~S*,*£ys»-(£*.y.)» 
i         i  i 

is  non-negative.  This  result,  which  was  known  to  Cauchy  and  Bessel,  is 
frequently  called  Schwarz's  inequality  as  Schwarz  obtained  a  similar 
inequality  for  integrals. 

*  L.  Fej£r,  Math.  Zeits.  Bd.  i,  S.  70  (1918). 


Hermitian  Forms  39 

Using  this  particular  form  of  h  in  Fejer's  inequality  we  obtain  the  result 
that 

n,  n  n  n 

Xgrsyry,<X9rrXy*. 

1,1  11 

For  further  properties  of  quadratic  forms  the  reader  is  referred  to  Brom- 
wich's  tract,  Quadratic  Forms,  Cambridge  (1906),  to  Bocher's  Algebra, 
Macmillan  and  Co.  (1907),  and  to  Dickson's  Modern  Algebraic  Theories, 
Sanborn  and  Co.,  Chicago  (1926). 

§  1-36.   Hermitian  forms.  Let  z  denote  the  complex  quantity  conjugate 
to  a  complex  quantity  z  and  let  the  complex  coefficients  crs  be  such  that 


n,n 

the  bilinear  form  2  crszrzs 

i,  i 

is  then  Hermitian,  If  clm  =  alm  +  iblm,zm  =  rmelQm,  where  alm,blm,  rm,  9m 
are  real  quantities,  we  have 

alm  ~  aml>       bim  =   —  Omi, 

and  the  Hermitian  form  can  be  expressed  as  a  quadratic  form 

n,n 

2  ptmrirm, 
i,  i 

where  plm  =  alm  cos  (0t  -  6J  +  blm  sin  (0t  -  0m)  =  pml. 

The  positive  definite  Hermitian  forms  which  are  positive  whenever  at 
least  one  of  the  quantities  zl  ,  z2  ,  .  .  .  zn  is  different  from  zero  are  of  special 
interest.  In  this  case  the  associated  quadratic  form  is  positive  for  all  non- 
vanishing  sets  of  values  of  rl  ,  r2  ,  .  .  .  rn  and  for  all  values  of  01  ,  62l  ...  6n. 
An  important  property  of  a  Hermitian  form  is  that  the  associated  secular 
equation 

|  crs  -  A8r.  |  =  0 

8r3  =  1      r  =  s 
-  0      r  +  s 

has  only  real  roots.  When  the  form  is  positive  these  roots  are  all  positive. 
The  proof  of  this  theorem  may  be  based  upon  analysis  very  similar  to  that 
given  in  §  1-33. 

EXAMPLES 
1.   If  F(t)  ^  0  for  -  TT  ^  *  ^  TT  and 


=    2    c¥eM, 

\>  =s  ~  00 

n,  n 
#n=    2    Cl-mZlZm          n»  1,2,3..., 

then  #n  ^  0.  This  has  been  shown  by  Carath£odory  and  Toeplitz  to  be  a  necessary  and 
sufficient  condition  that  F(t)  >  0.  See  Rend.  Palermo,  t.  xxxn,  pp.  191,  193  (1911). 


40  The  Classical  Equations 

2.   If  /(*)=  f°  eiteo>  (x)dx, 

J  —  oo 

where  o>  (a;)  =  to  (—  #) 

n,  n  _ 

ftnd  #„=   2  a)  (a?j  -  *m)  fzfm, 

Mathias  has  shown  that  when  f(t)  ^  0  we  have  Hn  ^  0  for  any  choice  of  real  parameters 
xl  ,  x2  ,  .  .  .  xn  and  of  the  complex  numbers  f,  ,  f,  ,  .  .  .  fn  .  See  Jlf  ath.  Zeite.  Bd.  xvi,  p.  103  (  1923). 
The  analysis  depends  upon  Fourier's  inversion  formula  which  is  studied  in  §  3-12  and  it 
appears  that,  with  suitable  restrictions  on  the  function  <*>(x),  the  inequality  Hn  ^>  0  is  the 
necessary  and  sufficient  condition  that  f(t)  ^  0.  Mathias  gives  two  methods  of  choosing  a 
function  a>(x)  which  will  make  Hn  ^  0.  The  correctness  of  these  should  be  verified  by  the 
reader. 

(1)  If  the  functions  x(x)>  x(x)  are  such  that  when  x  has  any  real  value  x(x)  an(l  X  (x)  are 
conjugate  complex  quantities,  the  function 

/«> 
X(a+  %)\(a  -  x)da 
-00 

will  make  Hn  ^  0. 

(2)  If  the  positive  constants  \v  and  the  functions  xv(x)  are  such  that 


is  a  function  of  x  —  y>  say  o>  (x  —  y)y  then  this  function  o>  (x)  will  make  Hn  ^  0. 

§  1-41.  Forced  oscillations.  When  a  particle,  which  is  normally  free  to 
oscillate  with  simple  harmonic  motion  about  a  position  of  equilibrium,  is 
acted  upon  by  a  periodic  force  varying  with  the  time  like  sin  (pt),  the 
equation  of  motion  takes  the  form 

x  4-  k2x  =  A  sin  pt. 

Writing  x  =  z  -f-  C  sin  pt,  where  G  is  a  constant  to  be  determined,  we 
find  that  if  we  choose  C  so  that 

(k*-p*)C  =  A,  ......  (A) 

the  equation  for  z  takes  the  form 

z  -f  k*z  =  0. 

The  motion  thus  consists  of  a  free  oscillation  superposed  on  an  oscilla- 
tion with  the  same  period  as  the  force.  In  other  words  the  motion  is  partly 
original  and  partly  imitation.  It  should  be  noticed,  however,  that  if  p*  >  k2 
the  imitation  is  not  perfect  because  there  is  a  difference  in  phase.  The 
difference  between  the  case  p*  >  k2  and  the  case  p2  <  k2  is  beautifully 
illustrated  by  giving  a  simple  harmonic  motion  to  the  point  of  suspension 
of  a  pendulum. 

When  p2  ==  k2  the  quantity  C  is  no  longer  determined  by  equation  (A)  and 
the  solution  is  best  obtained  by  the  method  of  integrating  factors  which 
may  be  applied  to  the  general  equation 

x  +  k2x  =  F  (t). 


Forced  Oscillations  41 

Multiplying  successively  by  the  integrating  factors  cos  kt  and  sin  kt  and 

integrating,  we  find  that  if  x  =  c,  x  =  u  when  t  =  T,  we  have 

ft 
x  cos  kt  -f  fc  sin  kt  =  u  cos  &T  -f  &c  sin  kr  +  \  F  (s)  cos  ks  .  d#, 

r< 
#  sin  kt  —  kx  cos  kt  —  u  sin  &r  —  kc  cos  &r  +  \  F  (s)  sin  &s  .  ds, 

rt 
kx  =  u  sin  A;  (t  —  r)  -f  &c  cos  k  (t  —  r)  -f      F  (s)  sin  &($  —  *)  ds, 

ct 
x  =  u  cos  A:  (t  —  r)  —  &c  sin  k  (t  —  r)  -f  I   F  (s)  cos  k  (t  —  s)  ds, 

J  T 

the  function  F  (s)  being  supposed  to  be  integrable  over  the  range  T  to  t. 

In  particular,  if  the  particle  starts  from  rest  at  the  time  t  we  have  at 
any  later  time  t 

kx  =  IF  (s)  sin  k  (t  —  s)  ds, 

rt 
x  =  IF  (s)  cos  k  (t  —  s)  ds. 

J  r 

When  F  (s)  =  A  sin  ks  and  r  =  0  we  find  that 
2fcr  =  t  cos  itf  —  &"1  sin  kt, 

and  the  oscillations  in  the  value  of  x  increase  in  magnitude  as  t  increases. 
This  is  a  simple  case  of  resonance,  a  phenomenon  which  is  of  considerable 
importance  in  acoustics.  In  engineering  one  important  result  of  resonance 
is  the  whirling  of  a  shaft  which  occurs  when  the  rate  of  rotation  has  a 
critical  value  corresponding  to  one  of  the  natural  frequencies  of  lateral 
vibration  of  the  shaft.  For  a  useful  discussion  of  vibration  problems  in 
engineering  the  reader  is  referred  to  a  recent  book  on  the  subject  by 
S.  Timoshenko,  D.  Van  Nostrand  Co.,  New  York  (1928). 

By  choosing  the  unit  of  time  so  that  k  =  1  the  mathematical  theory 
may  be  illustrated  geometrically  with  the  aid  of  the  curve  whose  radius 
of  curvature,  p,  is  given  by  the  equation  p  =  a  sin  OHJJ,  where  ^  is  the  angle 
which  the  tangent  makes  with  a  fixed  line.  Using  p  now  to  denote  the 
length  of  the  perpendicular  from  the  origin  to  the  tangent,  we  have  the 
equation  ^ 

......  (B) 


The  quantity  p  thus  represents  a  solution  of  the  differential  equation, 
and  by  suitably  choosing  the  position  of  the  origin  the  arbitrary  constants 

in  the  solution  can  be  given  any  assigned  real  values.   In  this  connection 

d*D 
it  should  be  noticed  that  ~j  has  a  simple  geometrical  meaning  (Fig.  7). 

When  co  =  1  the  equation  (B)  is  that  of  a  cycloid,  while  epicycloids  and 
hypocycloids  are  obtained  by  making  a)  different  from  unity.  The  intrinsic 
equation  of  these  curves  is  in  fact 

4  (a  -f  b)  b   .       aJj 

Q  —       V  '  „  am     .....  T  ___ 

o  —  -         OIJ.JL  rt,  , 

a  a  +  2b 


42  The  Classical  Equations 

where  a  is  the  radius  of  the  fixed  circle  and  b  the  radius  of  the  rolling  circle 
which  contains  the  generating  point. 

Epicycloids  6  >  0. 

Pericycloids  and  hypocycloids  6  <  0. 


Fig.  8. 

§  1-42.  The  effect  of  a  transient  force  in  producing  forced  oscillations 
is  best  studied  by  putting  r  =  —  oo  and  assuming  that  c  and  u  are  both 
zero,  then 

Too 

kx  =        F  (t  —  cr)  sin  ka  .  da, 

Jo 


f00 
x  =        F  (t  —  a)  cos  ka  .  da. 

' 


Let  us  consider  first  of  all  the  case  when 


In  this  case  F'  (t)  is  discontinuous  at  time  t  =  0  in  a  way  such  that 

F'  (-  0)  -  F'  (-f  0)  -  2h. 

The  solution  which  is  obtained  by  supposing  that  x,  x  and  x  are  con 
tinuous  at  time  t  =  0  is 


ft  Too 

lex  =      e-w-*>  sin  kada  -f 

.'0  J« 


2  A  sin  let 
k*+h* 


_ht 


.  _  2h  cos  kt  —  he~ht 

£  ==  ~v"o ;     »~  o ~ 


i  >  0 


It  will  be  noticed  that  x  is  discontinuous  at  t  =  0. 


Residual  Oscillation  43 

It  is  clear  from  these  equations  that  x  increases  with  t  until  t  reaches 
the  first  positive  root  of  the  transcendental  equation 

2  cos  kt  =  e~ht. 
The  corresponding  value  of  x  is  then 

2  (h  sin  kt  -f  k  cos  kt) 

As  t  increases  beyond  the  critical  value  x  begins  to  oscillate  in  value, 
and  as  t  ->  oo  there  is  an  undamped  residual  oscillation  given  by 

_  2h  sin  kt 

A  general  formula  for  the  displacement  x  at  time  t  in  the  residual 
oscillation  produced  by  a  transient  force  may  be  obtained  by  putting 
t  =  oo  in  the  upper  limit  of  the  integral  (t  being  retained  in  the  integrand)  *. 
This  gives 

kx  =  I      F  (s)  sin  k  (t  —  s)  ds. 

J    -00 

In  particular,  if 

,,  ,  x       [b  7         sin  bs 

F  (s)  =       cos  ms  .  dm  =    —  - , 

Jo  s 

we  have 

/*QO                     f?  Q  r  °^  ft  ^ 

kx  —  sin  kt        cos  ks  sin  bs cos  kt        sin  ks  sin  65      ; 

.'  -oo  S  J  _oo  S 

the  second  integral  vanishes  arid  we  may  write 

TOO  •  ^£ 

2&#  =  sin  kt        [sin  (ft  -f  k)  s  -f  sin  (6  —  k)  s]  — 

J  -oo  S 

=  277  sin  &£         if  6  >  &  >  0 
'  =  0  if  0  <  b  <  k 

=  TT  sin  kt  if  b  =  &  >  0. 

There  is  thus  a  residual  oscillation  only  when  b  >  k. 

EXAMPLES 

/•& 

1.  If  F  (s)  =  I ,   cos  m*  .  dm, 

there  is  a  residual  oscillation  only  when  k  lies  within  the  range  a  <  &<  6.  Extend  this  result 
by  considering  cases  when 

F  (8)  =  /    cos  ms .  <f>  (m)  dm,     F  (s)  =  /     sin  ms .  </>  (m)  dm, 
J  a  J  a 

^  (m)  being  a  suitable  arbitrary  function. 

2.  Determine  the  residual  oscillation  in  the  case  when 

F  (s)  -  (c2  -f  s2)-1. 
*  Cf.  H.  Lamb,  Dynamical  Theory  of  Sound,  p.  19. 


44  The  Classical  Equations 

3.  If  F  (s)  =  se~h  I s  I ,  where  h  >  0  and  #  is  chosen  to  be  a  solution  of  the  differential 
equation  and  the  supplementary  conditions  x  —  0,  x  —  0,  when  t  =  —  oo  ,  x  and  x  continuous 
at  t  —  0,  the  residual  oscillation  is  given  by 

4th  cos  kt 

X^~(k*~+h*)2' 
If  k2  >  h2  there  is  a  negative  value  of  t  for  which  r  —  0,  but  if  k2  <  h2  there  is  no  such  value. 

4.  If  .c  +  k*x  =  x<*(t)> 

where  o>(/)  is  zero  when  t~  oo  and  is  bounded  for  other  real  values  of  t,  the  solution  for  which 
x  and  b  are  initially  zero  can  be  regarded  as  the  residual  oscillation  of  a  simple  pendulum 
disturbed  by  a  transient  force. 

5.  If  0  <  a2  <  A  (t)  <  62,  the  differential  equation 

.-  +  A(t)x  =  T)  (C) 

is  satisfied  only  by  oscillating  functions.  Prove  that  the  interval  between  two  consecutive 
roots  of  the  equation  x  —  0  lies  between  rr/a  and  TT/&. 

[Let  y  be  a  solution  o£y  4-  62y  =  0  which  is  positive  in  the  interval  r±  <  t  <  r2,  then 


Let  us  now  suppose  that  it  is  possible  for  x  to  be  of  one  sign  (positive,  say)  in  the  interval 
T,  ^  t  <  T2  and  zero  at  the  ends  of  the  interval.  We  are  then  led  to  a  contradiction  because 
the  suppositions  make  .r  positive  near  ^  and  negative  near  r2,  they  thus  make  the  left-hand 
side  negative  (or  zero)  and  the  right-hand  side  positive.  Hence  the  interval  between  two 
consecutive  roots  of  x  must  be  greater  than  any  range  in  which  y  is  positive,  that  is,  greater 
than  y/b.  In  a  similar  way  it  may  be  shown  that  if  z  is  a  solution  of  z  -f  a2z  =  0  the  interval 
between  two  consecutive  roots  of  the  equation  z  —  0  is  greater  than  any  range  in  which  x  ^  0.] 

This  is  one  of  the  many  interesting  theorems  relating  to  the  oscillating  functions  which 
satisfy  an  equation  of  type  (C).  For  further  developments  the  reader  is  referred  to  Bocher's 
book,  Lemons  sur  les  methodes  de  Sturm,  Gauthier-Villars,  Paris  (1917)  and  his  article, 
" Boundary  problems  in  one  dimension,"  Fifth  International  Congress  of  Mathematicians, 
Proceedings,  vol.  I,  p.  163,  Cambridge  (1912). 

6.   Prove  that  x2  -f  x2  remains  bounded  as  t  ->  x  but  may  not  have  a  definite  limiting 
value,  x  being  any  solution  of  (C).   [M.  Fatou,  Compte*  Rendus,  t.  189,  p.  967  (1929).] 
The  generality  of  this  result  has  recently  been  questioned.  See  Note  I,  Appendix. 

§  1-43.  Motion  with  a  resistance  proportional  to  the  velocity.  Let  us  first 
of  all  discuss  the  motion  of  a  raindrop  or  solid  particle  which  falls  so  slowly 
that  the  resistance  to  its  motion  through  the  air  varies  as  the  first  power 
of  the  velocity.  This  is  called  Stokes'  law  of  resistance;  it  will  be  given  in 
a  precise  form  in  the  section  dealing  with  the  motion  of  a  sphere  through 
a  viscous  fluid ;  for  the  present  we  shall  use  simply  an  unknown  constant 
coefficient  k  and  shall  write  the  equation  of  motion  in  the  form 

mdv/dt  =  m'g  —  kv,  (A) 

where  m  is  the  apparent  mass  of  the  body  when  it  moves  in  air,  m'  is  the 
reduced  mass  when  the  buoyancy  of  the  air  is  taken  into  consideration  and 
g  is  the  acceleration  of  gravity.  The  solution  of  this  equation  is 

_kt 

kv  =  m'g  +  Be   m, 


Resisted  Motion  45 

where  -B  is  a  constant  depending  on  the  initial  conditions.   If  v  =  0  when 
t  =  0  we  have  tct 

kv  =  m'g(\  -  e    '*). 
To  find  the  distance  the  body  must  fall  to  acquire  a  specified  velocity 

v  we  write  v  =  x  ,  ,  ,  ,  7 

mvdv/dx  =  mg  —  kv, 


/    i 

'gr  log  -- 

y     &  Vra  0  -  kv 

If  T7  denote  the  terminal  velocity  the  equation  may  be  written  in  the 
form  v 

m'gx  =  ra  F2  log  -=%-  --      —  m  Vv. 

Let  us  now  consider  the  case  in  which  the  particle  moves  in  a  fluctuating 
vertical  current  of  air.  Let  /'  (t)  be  the  upward  velocity  of  the  air  at  time 
t,  v  the  velocity  of  the  particle  relative  to  the  ground  and  u  =  v  +  /x  (t)  the 
relative  velocity.  On  the  supposition  that  the  resistance  is  proportional  to 
the  relative  velocity  the  equation  of  motion  is 

mdv/dt  =  m'g  —  ku  ---  m'g  —  kv  ~  kf  (t). 

If  v  =  0  when  t  =  0  the  solution  is 


The  last  integral  may  be  written  in  the  form 


/'(t-r)e     "'dr, 

/  0 

which  is  very  useful  for  a  study  of  its  behaviour  when  t  is  very  large. 
The  distance  traversed  in  time  /  is  given  by  the  equation 


the  constant  in/  (t)  being  chosen  so  that/  (0)  =  0.   In  particular  if 

~~~k~  '  p' 

i  ki 

clc  — 

we  have  v  =  — «—- 0- — ,-*  [ke   m  —  mP  sin  pt  —  k  cos  pi] , 

m2p2  +  k2L  *        r  r  j» 

ck  *  — 

r  7  ^  vn~\ 


As  t  ->  oo  we  are  left  with  a  simple  harmonic  oscillation  which  is  not  in 
the  same  phase  as  the  air  current. 

It  should  be  emphasised  that  this  law  of  resistance  is  of  very  limited 
application  as  there  is  only  a  small  range  of  velocities  and  radius  of  particle 


46  The  Classical  Equations 

for  which  Stokes'  law  is  applicable.  It  should  be  mentioned  that  the  product 
of  the  radius  and  the  velocity  must  have  a  value  lying  in  a  certain  range 
if  the  law  is  to  be  valid. 

An  equation  similar  to  (A)  may  be  used  to  describe  the  course  of  a 
unimolecular  chemical  reaction  in  which  only  one  substance  is  being  trans- 
formed. If  the  initial  concentration  of  the  substance  is  a  and  at  time  t 
altogether  x  gram-molecules  of  the  substance  have  been  transformed,  the 
concentration  is  then  a  —  x  and  the  law  of  mass  action  gives 

dx      . 


the  coefficient  k  being  the  rate  of  transformation  of  unit  mass  of  the  substance. 
Simultaneous  equations  involving  only  the  first  derivatives  of  the 
variables  and  linear  combinations  of  these  variables  occur  in  the  theory 
of  consecutive  unimolecular  chemical  reactions.  If  at  the  end  of  time  t  the 
concentrations  of  the  substances  A,  B  and  C  are  x,  y  and  z  respectively 
and  the  reactions  are  represented  symbolically  by  the  equations 

A-+B,     B-+C, 
the  equations  governing  the  reactions  are 

dx/dt  =  —  &!#,     dy/dt  =  ^x  —  k2y,     dz/dt  =  k2y. 

A  more  general  system  of  linear  equations  of  this  type  occurs  in  the 
theory  of  radio-active  transformations.  Let  P0,  Ply  ...  Pn  represent  the 
amounts  of  the  substances  A0,  A1  ,  ...  An  present  at  time  t,  then  the  law  of 
mass  action  gives  ,p 

_°-      A  P 
dt  ~~~  **    OJ 


~^  —  ^n-lPn-l 

where  the  coefficients  As  are  constants.  In  my  book  on  differential  equations 
this  system  of  equations  is  golved  by  the  method  of  integrating  factors. 
This  method  is  elementary  but  there  is  another  method*  which,  though 
more  recondite,  is  more  convenient  to  use. 

Let  us  write  ps  (x)  =  f  V*'P8  (t)  dt,  (B) 

J  Q 
Too  fjp 

then  e-*«  ~8  dt  =  -  P8  (0)  +  xps  (x), 

Jo  dt 

*  H.  Bateman,  Proc.  Camb.  Phil.  Soc.  vol.  xv,  p.  423  (1910). 


Reduction  to  Algebraic  Equations  47 

and  so  the  system  of  differential  equations  gives  rise  to  the  system  of 
linear  algebraic  equations 

xp,  (x)  -  P0  (0)  =  -  A^  (x), 

xpl  (x)  -  Pl  (0)  =  AO^O  (x)  ~  \pl  (x), 

xp2  (x)  -  P2  (0)  =  Aj^  (x)  -  X2p2  (x), 


Xpn  (X)  -  Pn  (0)  =  V,1>«-1  (X)  - 

from  which  the  functions  pQ  (x),  pt  (x),  ...  pn  (x)  may  at  once  be  derived. 

If  Pl  (0)  =  P2  (0)  =  ...  =  Ptt  (0)  =  0,  i.e.  if  there  is  only  one  substance 
initially,  and  if  P0  (0)  =  Q,  we  have 

p» (x)  =  x  TV   Pl  (x}  =  ~(*TAo)7aTXj ' 


To  derive  Ps  (t)  from  j9s  (x)  we  simply  express  ps  (x)  in  partial  fractions 

p  te)  =  -   C°  ____  h  ...  -     - 

X    ~T"     AQ  X    ~\-     Ag 

The  corresponding  function  Ps  (t)  is  then  given  by 


for  this  is  evidently  of  the  correct  form  and  the  solution  of  the  system  of 
differential  equations  is  unique.  The  uniqueness  of  the  function  Ps  (t) 
corresponding  to  a  given  function  ps  (x)  can  also  be  inferred  from  Lerch's 
theorem  which  will  be  proved  in  §  6-29. 

If  some  of  the  quantities  Aw  are  equal  there  may  be  terms  of  type 


in  the  representation  of  ps  (x)  in  partial  fractions.   In  this  case  the  corre- 
sponding term  in  Ps  (t)  is  n  _A  t 

^m,*  t  <  e      "t  . 

Such  a  case  arises  in  the  discussion  of  a  system  of  linear  differential 
equations  occurring  in  the  theory  of  probability*. 

§  1-44.  The  equation  of  damped  vibrations.  A  mechanical  system  with 
one  degree  of  freedom  may  be  represented  at  time  t  by  a  single  point  P 
which  moves  along  the  x-axis  and  has  a  position  specified  at  this  instant 
by  the  co-ordinate  x.  This  point  P,  which  may  be  called  the  image  of  the 
system,  may  in  some  cases  be  a  special  point  of  the  system,  provided  that 
the  path  of  such  a  point  is  to  a  sufficient  approximation  rectilinear.  The 

*  H.  Bateman,  Differential  Equations,  p.  45. 


48  The  Classical  Equations 

mechanical  system  may  also  in  special  cases  be  just  one  part  of  a  larger 
system ;  it  may,  for  instance,  be  one  element  of  a  string  or  vibrating  body 
on  which  attention  is  focussed.  To  obtain  a  simple  picture  of  our  system 
and  to  fix  ideas  we  shall  suppose  that  P  is  the  centre  of  mass  of  a  pendulum 
which  swings  in  a  resisting  medium. 

The  motion  of  the  point  P  is  then  similar  to  that  of  a  particle  acted  upon 
by  forces  which  depend  in  value  on  t,  x,  x  and  possibly  higher  derivatives. 
For  simplicity  we  shall  consider  the  case  in  which  the  force  F  is  a  linear 

function  of  x  and  x,      ™      /.  m       ,  m          07  u.  . 

F  =  /  (t)  —  h  (t)  x  —  2k  (t)  x. 

In  the  case  when  h  (t)  and  k  (t)  are  constants  the  equation  of  motion 

takes  the  simple  form                rt7  .         9         » /jt.  ,  A  v 

*  x  +  2kx  +  n2x  =  f  (t).  (A) 

The  motion  of  the  particle  is  in  this  case  retarded  by  a  frictional  force 
proportional  to  the  velocity.  If  it  were  not  for  this  resistance  the  free 
motion  of  the  particle  would  be  a  simple  harmonic  vibration  of  frequency 
ft/277.  The  effect  of  the  resistance  when  n2  >  k2  is  to  reduce  the  free  motion 
to  a  damped  oscillation  of  type 

x  =  Ae~kt  sin  (pt  +  e),  (B) 

where  A  and  €  are  arbitrary  constants  and 

pz  =  n2  -  k2. 

The  period  of  this  damped  oscillation  may  be  defined  as  the  interval 
between  successive  instants  at  which  x  is  a  maximum  and  is  %TT/P.  One 
effect  of  the  resistance,  then,  is  to  lengthen  the  period  of  free  oscillations. 
It  may  be  noted  that  the  interval  between  successive  instants  at  which 
x  =  0  is  2-rr/p.  The  time  range  0  <  t  <  oo  may,  then,  be  divided  up  into 
intervals  of  this  length.  The  sign  of  x  changes  as  t  passes  from  one  interval 
to  the  next  and  so  the  point  P  does  in  fact  oscillate.  Points  P  and  P'  of 
two  intervals  in  which  x  has  the  same  sign  may  be  said  to  correspond  if 
their  associated  times  t,  t',  are  connected  by  the  relation 

pt'  =  pt  +  2m7r, 
where  m  is  an  integer.  We  then  have 

x'  =  xe~k(t'~t}  =  xe~2kmirlp. 

The  positive  constant  k  is  seen,  then,  to  determine  the  rate  of  decay 
of  the  oscillations. 

When  n2  =  k2  the  free  motion  is  given  by 

x  =  (A  +  Bt)  e~kt, 

where  A  and  J5  are  arbitrary  constants.  In  this  case  x  vanishes  at  a  time 
t  given  by  B  =  k  (A  -f-  Hi),  thus  |  x  \  increases  to  a  maximum  value  and 
then  decreases  rapidly  to  zero.  The  motion  of  a  dead-beat  galvanometer 
needle  may  be  represented  by  an  equation  of  type  (A)  with  n2  =  k2. 


Damped  Vibrations  49 

When  k2  >  n2  the  free  motion  is  of  type 

x  =  Ae~ut  -f  fie-"', 
where  ^  and  v  are  the  roots  of  the  equation 

22  ~   2A'Z  +  7l2  -  0. 

In  this  ease  the  general  value  of  x  is  obtained  by  the  addition  of  two 
terms  each  of  which  represents  a  simple  subsidence,  the  logarithmic  de- 
crement of  which  is  u  for  the  first  and  v  for  the  second.  The  time  r  which 
is  needed  for  the  value  of  x  in  one  of  these  subsidences  to  fall  to  half  value 

is  given  by  the  equation  ^  __ 

2  —  e  UT- 

In  the  case  of  the  damped  oscillation  (B)  the  quantity  Ae~ki  can  be 
regarded  as  the  amplitude  at  time  t.  In  an  interval  of  time  r  this  diminishes 
in  the  ratio  r:  1,  where  r  =  ekr. 

Putting  T  =  1,  we  have  k  --  log  r,  whence  the  name  logarithmic  decre- 
ment usually  given  to  k.  Instead  of  considering  the  logarithmic  decrement 
per  unit  time  we  may,  in  the  case  of  a  damped  vibration,  consider  the 
logarithmic  decrement  per  period  or  per  half-period*.  It  is  knjp  for  the 
half  -period. 

When  /  (t)  —  C  sin  mt,  where  C  and  m  are  constants,  the  solution  of 
the  differential  equation  (A)  is  composed  of  a  particular  integral  of  type 

~  (n2  —  m2)  sin  nit  —  2km  cos  mt 

X    ~     C  .        0  0\0~     ,         A  1   9          "«T"  ......  (I) 

(n*  —  m2)2  -f  4:k~m2  v  ' 

and  a  complementary  function  of  type  (B).  The  particular  integral  is 
obtained  most  conveniently  by  the  symbolical  method  in  which  we  write 
D  =  djdt  and  make  use  of  the  fact  that  D2f  —  —  m2/.  The  operator  D  is 
treated  as  an  algebraic  quantity  in  some  of  the  steps 


____ 

-f-  2A-/>  H-  n2  ~  ^2  -  m2  -f  2k 
2-m2--  2kD 


_  __  2^  (  '' 

If  .r  =  0,  a;  =  0  when  ^  =  a  the  unknown  constants  in  the  complementary 
function  may  be  determined  and  we  find  that 

-  r)f(r)dr.  ......  (C) 

This  result  may  be  obtained  directly  from  the  differential  equation  by 
using  the  integrating  factors  ekt  sin  pt  and  eu  cos  pt. 
We  thus  obtain  the  equations 

px  (*<  sin  pt  +  (x  +  kx)  «*'  cos  p<  =  |    e*r  cos  pr  ./  (r)  dr> 


cos 


fc-r)  ^'  sin  p<  =      ekr  sin  pr  ./  (T) 


*  E.  H.  Barton  and  E.  M.  Browning,  Phil.  Mag.  (6),  vol.  XLVII,  p.  495  (1924). 
B  4 


50  The  Classical  Equations 

from  which  (C)  is  immediately  derived.   We  also  obtain  the  formula 

X  +  kx    -    [   e~k(t-T)COSp  (t  -  r)f(r)  dr. 

introducing  an  angle  c  defined  by  the  equation 

2km 

tan  e=2         2 
n2  —  m2 

the  particular  integral  (I)  may  be  expressed  in  the  form 

x  ~  A  siri  (mi  —  e), 
where  the  amplitude  A  is  given  by  the  formula 

A2  [(n2  ~  m2)2  +  4k2m2]  -  C2. 

N 

This  is  the  forced  oscillation  which  remains  behind  when  the  time  t  is  so 
large  that  the  free  oscillations  have  died  down.  The  amplitude  A  is  a 
maximum  when  m  is  such  that  m2  ^  n2  —  2k2  =-  p2  —  k2.  We  then  have 

^4max=  C/2kp. 

Writing  A  =  a/lmax  it  is  easily  seen  that  when  m  is  nearly  equal  to  n 
and  k/n  is  so  small  that  its  square  can  be  neglected  we  have  the  approxi- 
mate formula*  7  ,  ,  ,,  „.  i 

k  =  a  I  n  —  m  \  (1  —  a2)  *. 

This  formula  has  been  used  to  determine  the  damping  of  forced  oscillations 
of  a  steel  piano  wire. 

It  should  be  noticed  that  a  differential  equation  of  type  (A)  may  be 
obtained  from  the  pair  of  equations 

u  -f  ku  -f  nv  —  0, 
v  -f  kp  —  nu  =  0, 

where  k  and  n  are  constants.  These  represent  the  equations  of  horizontal 
motion  of  a  particle  under  the  influence  of  the  deflecting  force  of  the  earth's 
rotation  and  a  frictional  force  proportional  to  the  velocity.  These  equations 

k  uu  -f-  vv  -f  k  (u2  -f  v 2)  =  0, 

u2  +  v2  =  q2e-2kt, 

hence  k  is  the  logarithmic  decrement  for  the  velocity. 

The  equation  of  damped  vibrations  has  some  interesting  applications 
in  seismology  and  in  fact  in  any  experimental  work  in  which  the  motion 
of  the  arms  of  a  balance  is  recorded  mechanically. 

The  motion  of  a  horizontal  or  vertical  seismograph  subjected  to  dis- 
placements of  the  ground  in  a  given  direction,  say  x  —  f  (t),  can  be  repre- 
sented by  an  equation  of  form 

6  +  2kti  +  n20  +  x/l  =  0, 

*  Florence  M.  Chambers,  Phil  Mag.  (6),  vol.  XLVIII,  p.  636  (1924). 


Instrumental  Records  51 

where  6  is  the  deviation  of  the  instrument,  k  a  constant  which  depends 
upon  the  type  of  damping  and  I  the  reduced  pendulum  length*. 

The  motion  of  a  dead-beat  galvanometer,  coupled  with  the  seismograph, 
is  governed  by  an  equation  of  type 

$  +  2m(j>  -f  mty  +  h9  =  0, 

where  </>  is  the  angle  of  deviation  of  the  galvanometer  and  h  and  m  are 
constants  of  the  instrument. 

To  get  rid  as  soon  as  possible  of  the  natural  oscillations  of  the  pendulum, 
introduced  by  the  initial  circumstances,  it  is  advisable  to  augment  the 
damping  of  the  instrument,  driving  it  if  possible  to  the  limit  of  a  periodicity 
and  making  it  dead  beat.  By  doing  this  a  more  truthful  record  of  the  move- 
ment of  the  ground  is  obtained. 

When  n2  =  k2  the  solution  of  the  equation  of  forced  motion 

x  -f  2kx  +  k*x  -  /  (t) 
is  a=    *  (t-  T)e-k(i-r)f(T)dr^   (A  +  Bt)e~u. 

When  t  is  large  the  second  term  is  negligible  and  the  lower  limit  of  the 
integral  may,  to  a  close  approximation,  be  replaced  by  0,  —  oo,  or  any 
other  instant  from  which  the  value  of/  (t)  is  known. 

When  k  is  large  the  second  term  is  negligible  even  when  t  has  moderate 
values  and  if,  for  such  values  of  t,  f  (t)  is  represented  over  a  certain  range 
with  considerable  accuracy  by  C  sin  mt,  the  value  of  x  is  given  approxi- 
mately by  the  formula 

f(t)  _        Csinmt 


_ 
X  ~  D*  +  2kJ)  +  T2  "  k2  -  m2  +  2kD 


...(I') 


When  k  is  large  in  comparison  with  m  a  good  approximation  is  given  by 

k*x  =-  C  sin  mt  =/(/)> 

and  the  factor  of  proportionality  k2  is  independent  of  m,  consequently, 
if  a  number  of  terms  were  required  to  give  a  good  representation  of  /  (t) 
within  a  desired  range  of  values  of  t,  the  record  of  the  instrument  would 
still  give  a  faithful  representation,  on  a  certain  definite  scale,  of  the 
variation  of  the  force. 

When  k  and  an  are  of  the  same  order  of  magnitude  this  is  no  longer  true, 
consequently,  if  the  "high  harmonics"  occur  to  a  marked  degree  in  the 
representation  of/  (t)  by  a  series  of  sine  functions,  the  record  of  the  instru- 
ment may  not  be  a  true  picture  of  the  forcef. 

*  B.  Galitzm,  "The  principles  of  instrumental  seismology,"  Fifth  International  Congress 
of  Mathematicians,  Proceedings,  vol.  I,  p.  109  (Cambridge,  1912). 

f  If  m  =  &/10  the  solution  of  the  differential  equation  is  approximately  k2x  —  -99  sin  (mt  -  c), 
where  c  ia  the  circular  measure  of  an  angle  of  about  16°  59'.  When  m  =  k/5  the  solution  is  approxi- 
mately kzx  —  «96  sin  (mt  —  c),  where  e  is  the  circular  measure  of  an  angle  of  about  31°  47'. 


52  The  Classical  Equations 

When  n  ^  k  the  formula  (I')  shows  that  if  k  is  large  in  comparison  with 
both  m  and  n  the  solution  is  given  approximately  by  the  formula 

2kmx  =  ~  C  cos  mt 
which  may  be  written  in  the  form 


This  result  may  be  obtained  directly  from  the  differential  equation  by 
neglecting  the  terms  x  and  n2x  in  comparison  with  2kx.  In  this  case  the 
velocity  x  gives  a  faithful  record  of  the  force  on  a  certain  scale. 

Finally,  if  n2  is  large  in  comparison  with  k  and  ra,  formula  (I)  gives 


and  the  instrument  gives  a  faithful  record  of  the  force  when  the  natural 
vibrations  have  died  down. 

§  1-45.    The  dissipation  function.  The  equation  of  damped  vibrations 

mx  +  kx  +  {JLX  =  f  (t) 
may  be  written  in  the  form  of  a  Lagrangian  equation  of  motion 

d    dT\      dF     3V 


where  T  -  \mx\     F  -  \kx\     V  -  \^x\ 

Regarding  m  as  the  mass  of  a  particle  whose  displacement  at  time  t  is 
x,  T  may  be  regarded  as  the  kinetic  energy,  V  as  the  potential  energy  and 
F  as  the  dissipation  function  introduced  by  the  late  Lord  Rayleigh*.  The. 
function  F  is  defined  for  a  system  containing  a  number  of  particles  by  an 
equation  of  type  p  =  ^  (/^2  +  ^,  +  ^ 

where  kx,  ky,  kz  are  the  coefficients  of  friction,  parallel  to  the  axes,  for  the 
particle  x,y,z.  Transforming  to  general  co-ordinates  #1  ,  </2  >  <7a  >  .  .  .  qn  we  may 

write 


where  the  coefficients  [rs],  (rs),  {rs}  are  of  such  a  nature  that  the  quadratic 
forms  T,  F,  V  are  essentially  positive,  or  rather,  never  negative.  These 
coefficients  are  generally  functions  of  the  co-ordinates  ql9  ...  qn  ,  but  if  we 
are  interested  only  in  small  oscillations  we  may  regard  ql  ,  ...  qn  ,  ql9  ...  qn 
as  small  quantities  and  in  the  expansions  of  the  coefficients  in  ascending 
powers  of  ql9  ...  qn  it  will  be  necessary  only  to  retain  the  constant  terms 
if  we  agree  to  neglect  terms  of  the  third  and  higher  orders  in  (ft,  ...  qn, 
4i>  •••  ?«• 

*  Proc.  London  Math..Soc.  (1),  vol.  iv,  p.  357  (1873). 


Reciprocal  Relations  53 

The  generalised  Lagrangian  equations  of  motion  are  now 


- 

dt\dq 

where  Qm  is  the  generalised  force  associated  with  the  co-ordinate  qm.  Since 
T  is  supposed  to  be  approximately  independent  of  the  quantities  qlt 
q2,  ...  qn  the  second  term  may  be  omitted. 

Using  rs  as  an  abbreviation  for  the  quadratic  operator 


the  equations  of  motion  assume  the  linear  form 

H&  +  T2gr2+  ...  =  Q19 
2l?1+  22g2+  ...  =  Q2.  ......  (E) 

Since  [rs]  =  [sr],  (rs)  =  (sr),  (rs}  =  {sr},  it  follows  that  7s  =  sr. 

§  1-48.    Rayleigh's  reciprocal  theorem.   Let  a  periodic  force  Qs  equal  to 
As  cos  pt  act  on  our  mechanical  system  and  produce  a  forced  vibration  of 


=  KA8  cos  (pi  -  e), 

where  X  is  the  coefficient  of  amplitude  and  e  the  retardation  of  phase.  The 
reciprocal  theorem  asserts  that  if  the  system  be  acted  on  by  the  force 
Qr  =  A  cos  pi,  the  corresponding  forced  vibration  for  the  co-ordinate  </s, 

will  be  v  .          . 

qs  =  KAr  cos  (pt  —  €). 

Let  D  denote  the  determinant 

IT     12  13  .. 

21     22  23  .. 

31     32  33  .. 


and  let  rs  denote  its  partial  derivative  with  respect  to  the  constituent  rs 
when  no  recognition  is  made  of  the  relation  rs  =  sr  and  when  all  the 
constituents  are  treated  as  algebraic  quantities.  This  means  that  rs  is  the 
cofactor  of  rs  in  the  determinant  operator  D. 

Solving  the  equations  (E)  like  a  set  of  linear  algebraic  equations  on 
the  assumption  that  D  ^  0,  we  obtain  the  relations 

f  ...nlQn, 
...n2Qn, 


From  a  property  of  determinants  we  may  conclude  that  since  rs  =  sr 
we  have  also  rs  =  sr .  Thus  the  component  displacement  qr  due  to  a  force 
<?.  is  given  by  - 


54  The  Classical  Equations 

Similarly,  the  component  displacement  qs  due  to  a  force  Qr  is  given  by 

Dq8  =  srQr. 
Distinguishing  the  second  case  by  adash  affixed  to  the  various  quantities, 


where  the  coefficients  As,  A/  may  without  loss  of  generality  be  supposed 
to  be  real.    If  they  were  complex  but  had  a  real  ratio  they  could  be  made 
real  by  changing  the  initial  time  from  which  t  is  measured. 
Expressing  the  solution  in  the  form 

T  W  S7* 

0    —    A     "5  pu,t        n  '  —   A  Clpt 

(Jr  —  ^1  8  j~j  V      ,       ys    —  sir   jpG      , 

and  defining  the  forced  vibration  as  the  particular  integral  obtained  by 
replacing  d/dt  in  each  of  the  operators  by  ip,  we  obtain  the  relation 

A/qr  =  Asq; 
which  gives  reciprocal  relations  for  both  amplitude  and  phase. 

In  the  statical  case  the  quantities  [rs],  (rs),  are  all  zero  and  D,  rs,  rs 
are  simply  constants.  Rayleigh  then  gives  two  additional  theorems 
corresponding  to  those  already  considered  in  §  2. 

(2)    Suppose  that  only  two  forces  Ql3  Q2  act,  then 


If  ql  =  0  we  have      ^  ^ 

2  =  [122- 


From  this  we  conclude  that  if  q2  is  given  an  assigned  value  a  it  requires 
the  same  force  to  keep  q^  =  0  as  would  be  required  if  the  force  Q2  is  to  keep 
q2  =  0  when  ql  has  the  assigned  value  a. 

(3)    Suppose,  first,  that  Q1  ==•  0,  then  the  equations  (F)  give 

?1:?8=21:22. 

Secondly,  suppose  q2  =  0,  then 

02:0!=  -12:22. 

Thus,  when  Q2  acts  alone,  the  ratio  of  the  displacements  ql9  q2  is  —  Q^IQu 
where  Q11  Q%  are  the  forces  necessary  to  keep  q2  —  0. 

§  1-47.  Fundamental  equations  of  electric  circuit  theory.  A  system  of 
equations  analogous  to  the  system  (E)  occurs  in  the  theory  of  electric 
circuits.  This  theory  may  be  based  on  KirchhofTs  laws. 

(1)  The  total  impressed  electromotive  force  (E.M.P.)  taken  around  any 
closed  circuit  in  a  network  is  equal  to  the  drop  of  electric  potential  ex- 
pressed as  the  sum  of  three  parts  due  respectively  to  resistance,  induction 
and  capacity. 


Electric  Circuits  55 

Thus,  if  we  consider  an  elementary  circuit  consisting  of  a  resistance 
element  of  resistance  jR,  an  inductance  element  of  self-induction  (or  in- 
ductance) L  and  a  capacity  element  of  capacity  (capacitance)  (7,  all  in 
series,  and  suppose  that  an  E.M.F.  of  amount  E  is  applied  to  the  circuit, 
Kirchhoff's  first  law  states  that  at  any  instant  of  time 

RI  +  LTt  +  9C  =  E>  W 

where  /  is  the  current  in  the  circuit  and  Q  =  \Idt. 

The  fall  of  potential  due  to  resistance  is  in  fact  represented  by  KI 
where  R  is  the  resistance  of  the  circuit  (Ohm's  law),  the  drop  due  to  in- 
ductance is  Ldl/dt  and  the  drop  across  the  condenser  is  Q/C. 

There  is  an  associated  energy  equation 


in  which  RI  represents  the  rate  at  which  electrical  energy  is  being  converted 
into  heat,  while  the  second  and  third  terms  represent  rates  of  increase  of 
magnetic  energy  and  electrical  energy  respectively.  The  right-hand  side 
represents  the  rate  at  which  the  impressed  E.M.F.  is  delivering  energy  to 
the  circuit,  while  the  left-hand  side  is  the  rate  at  which  energy  is  being 
absorbed  by  the  circuit.  The  inductance  element  and  the  condenser  may 
be  regarded  as  devices  for  storing  energy,  while  the  resistance  is  responsible 
for  a  dissipation  of  energy  since  the  energy  converted  into  Joule's  heat  is 
eventually  lost  by  conduction  and  radiation  of  heat  or  by  conduction  and 
convection  if  the  circuit  is  in  a  moving  medium. 

If  we  regard  Q  as  a  generalised  co-ordinate,  we  may  obtain  the  equation 
(I)  by  writing  y  =  ^  f  _  ^  V  =  Q2/2C> 

dF     dV      _, 
.  -  .  f     .-  =  E. 

dQ      vQ 

(2)  In  the  case  of  a  network  the  sum  of  the  currents  entering  any 
branch  point  in  the  network  is  always  zero. 

If  we  consider  a  general  form  of  network  possessing  n  independent 
circuits,  Kirchhoff's  second  law  leads  to  the  system  of  equations 

-Er,  (II) 


(r=  1,2,  ...n), 

where  Er  is  the  E.M.F.  applied  to  the  rth  circuit,  Lss,  R9SJ  CKS  denote  the 
total  inductance,  resistance  and  capacitance  in  series  in  the  circuit  s, 
while  Lrs,  Rrs,  Crs  denote  the  corresponding  mutual  elements  between 
circuits  r  and  s.  We  have  written  Frs  for  the  reciprocal  of  Crs  and  Qs  for 

I/, eft,  where  Is  is  the  current  in  the  5th  circuit  or  mesh. 


56  The  Classical  Equations 

An  elaborate  study  of  these  equations  in  connection  with  the  modern 
applications  to  electrotechnics  has  been  made  by  J.  R.  Carson*. 

In  discussing  complicated  systems  of  resistances,  inductances  and 
capacities  it  will  be  convenient  to  use  the  symbol  (LRC)  for  an  inductance 
L,  a  resistance  R  and  a  capacity  C  in  series.  If  a  transmission  line  running 
between  two  terminals  T  and  T'  divides  into  two  branches,  one  of  which 
contains  (LRC)  and  the  other  (L'R'C'},  and  if  the  two  branches  subse- 
quently reunite  before  the  terminal  T'  is  reached,  the  arrangement  will  be 
represented  symbolically  by  the  scheme 


In  electrotechnics  a  mechanical  system  with  a  period  constant  n  and 
a  damping  constant  k  is  frequently  used  as  the  medium  between  the 
quantity  to  be  studied  (which  actuates  the  oscillograph)  and  the  record. 
The  oscillograph  is  usually  critically  damped  (k  =  n)  so  as  to  give  a  faithful 
record  over  a  limited  range  of  frequencies  but  even  then  the  usefulness  of 
the  instrument  is  very  limited  as  the  range  for  accurate  results  is  given 
roughly  by  the  inequality  10m  <  n. 

A  method  of  increasing  the  working  range  of  such  an  oscillograph  has 
been  devised  recently  by  Wynn-  Williams  f.  If  an  E.M.F.  of  amount  E  acts 
between  two  terminals  T  and  T'  and  the  aim  is  to  determine  the  variation 
of  E,  the  usual  plan  is  to  place  the  oscillograph  0  in  line  with  T  and  T'  so 
that  T,  0  and  T'  are  in  series.  In  our  notation  the  arrangement  is 

T  -  0  -  T. 
Instead  of  this  Wynn-  Williams  proposes  the  following  scheme 


in  which  L,  R  and  C  are  chosen  so  that  2k  =  JR/L,  n2  =  l/CL  and  L19  K19  C1 
are  chosen  so  that  Ll,  C\  are  small  and  R1  is  such  that  there  is  a  relation 
(k  4-  ^)2  -  n2  +  n^9  where  2k,  -  J?,//,,  nf  =  l/LC^ 

Putting  L)  =  0  and  writing  q  for  the  current  flowing  between  T  and 
T'  ,  x  for  the  reading  of  the  oscillograph,  F  for  the  back  E.M.F.  of  the  system 
(LRC),  we  have 


Therefore  x  =  -~; 


c\ 

[D2  +  2kD  +  n2]  x  =  F. 
E 


*  Klcctnc  circuit  theory  and  operational  calculus  (McGraw  Hill,  1926). 
f  Phil.  Mag.  vol.  L,  p.  1  (1925). 


Cauchy's  Method  57 

Hence  when  the  reading  of  the  oscillograph  is  used  to  determine  E  the 
oscillograph  behaves  as  if  its  period  constant  were  (n2  -f  n^  instead  of 
n  and  its  damping  constant  k  -f  1cl  instead  of  k.  By  choosing  n^  =  24n2, 
&x  =  4k  =  4n  we  have 

N  =  (n2  +  ft!2)*  =  5n,     Ar  -  k  +  ^  -  5fc  -  5w, 

thus  we  obtain  an  amplitude  scale  which  is  the  same  as  that  of  an  oscillo- 
graph with  a  period  constant  5n  and  a  damping  constant  5  A1. 

§  1-48.    Cauchy^s  method  of  solving  a  linear  equation*.    Let  us  suppose 
that  we  need  a  particular  solution  of  the  linear  differential  equation 

<f>(D)u  =  f(t),  ......  (I) 

where  (f>  (D)  =  a0i>  +  ^D"-1  4-  ...  an, 

and  D  denotes  the  operator  djdt.  The  coefficients  #s  are  either  constants  or 
functions  of  t.  For  convenience  we  shall  write  a  (t)  =  l/a0. 

If  (v1}  v2,  ...  #n)  are  distinct  solutions  of  the  homogeneous  equation 


the  coefficients  Cl  ,  C2  ,  .  .  .  Cn  in  the  general  solution 

t>  =   ^1^1  -f   ^2^2+    •••   GnVn 

may  be  chosen  so  that  v  satisfies  the  initial  conditions 

v  (r)  =  v'  (r)  =  ...  ^^~2>  (r)  -  0,      v^-1*  (T)  =  a  (T)/  (T), 

where  i;(s)  (£)  denotes  the  ,sth  derivative  of  v  (t)  and  r  is  the  initial  value  of 
t.  We  denote  this  solution  by  the  symbol  v  (t,  r)  and  consider  the  integral 


u  (t)  -  [  v  (t,  r)  dr. 

Jo 


Assuming  that  the  differentiations  under  the  integral  sign  can  be  made 
by  the  rule  of  Leibnitz,  we  have 


rt 
=\ 

Jo 


=  1,2,  ...n-  2,n- 


the  terms  arising  from  the  upper  limit  vanishing  on  account  of  the  pro- 
perties of  the  function  v.   On  the  other  hand 


Dnu  =       D"v  (t,  r)dr-\-a  (t)f(t), 

Jo 

and  so  <f>  (D)  u  =  /  (t)  +  f  V  (D)  v  .  dr  =  /  (t). 

Jo 

This  particular  solution  is  characterised  by  the  properties 

u  (0)  -  u'  (0)  =  ...  tt<"-i>  (0)  -  0,     w(n)  (0)  =  a  (O)/  (0). 
If,  when  /  (0  =  1>  ^  (^  T)  ==  e/r  (£,  r),  the  general  value  for  an  arbitrary 

*  Except  for  some  slight  modifications  this  presentation  follows  that  of  F.  D.  Murnaghan, 
Bull.  Amer.  Math.  Soc.  vol.  xxxm,  p.  81  (1927). 


58  The  Classical  Equations 

function  /  (t)  is  seen  to  be  /  (t)  if/  (t,  T)  and  so  we  may  write  u  (t)  in  the 
form  t 

u(t)=\    t(t,T)f(t)dr.  ......  (II) 

Jo 

Introducing  the  notation 


=  J 


we  have  -g—  £  (t,  r)  =  -  iff  (t,  T), 

and  the  integral  in  (II)  may  be  integrated  by  parts  giving 


u  (t)  =/(0)  f  (t,  0)  +       f  (*,  r)/'  (r)  dr. 

Jo 

This  is  the  mathematical  statement  of  the  Boltzmann-Hopkinson 
principle  of  superposition,  according  to  which  we  are  able  to  build  up  a 
particular  solution  of  equation  (I)  from  a  corresponding  particular  solution 
£  (t,  r)  for  the  case  in  which 

/W=l        t>*> 
=    0        t<    T. 

When  the  coefficients  in  the  polynomial  </>  (D)  are  all  constants  we  may 
Write  a0  (D  -  rt)  (D  -  r,)  .  .  .  (Z>  -  rj, 


where  r1?  r2,  ...  rn  are  the  roots  of  the  algebraic  equation  <f>  (x)  =  0.  Taking 
first  the  case  in  which  these  roots  are  all  distinct,  we  write 

vs  =  er»«-T>     s  =  1,  2,  ...  n. 

The  equations  to  determine  the  constants  C  are  then 


We  may  solve  for  C  by  multiplying  these  equations  respectively  by 
the  coefficients  of  the  successive  powers  of  x  in  the  expansion 

(/y*  — —    V   \    ( *Y   —    V   \  If        ,    ff    \    • —    /j      - 1 ,     r)    <•>»      I  h  /j»W — 1 

x  —  r2)  (x  —  r3) ...  (x  —  rn)  —  o0  -t-  u^x  -t-  ...  vn-iX 

Since  bn_t  =  1  we  find  that 

«/  (^  =  Ci  (T!  -  r2)  (r±  -  r3) ...  fa  -  rn)  =  aCtf  fa), 
and  the  other  coefficients  may  be  determined  in  a  similar  way. 
Writing  kr  for  the  reciprocal  of  <£'  (rs)  we  have 

00 

v  (f    -\  _   f  /-\     V     ^  ^^(i-r)  _   /"  /-.\  ./.  //     -\ 
v  \l'>  T/  —  y  VT/    ^    ^se  —  7  \T/  T  \^y  T)> 

8-1 

and  it  ps  denotes  the  reciprocal  of  r8 

n 

g  (t,  r)  =   2  paka  [ers(i~r)  —  1]. 

a-l 


Heaviside's  Expansion  59 

i          n      if 


and  so  [<£  (O)]-1  -  -   S  p^, 


When  there  is  a  double  root  rl  =  r2,  we  write 

^  =  eri«-T>,     v2  -  (t  -  T)  efi«-T), 

v3  ,  ...  vn  being  the  same  as  before.  The  equations  to  determine  the  constants 
Carenow  <74  +  0  .  <7,  +  C7,  +  ...  0.  =  0, 

r^  +  1  .  <72  +  r3  .  (73  +  ...  rn  .  Cn  =  0, 
/y-i/7,  +  (»-!)  r.-^C,  +  rj-'C1,  +  ...  r,,--^,  =  af  (t). 
Writing  (r  -  A)  F  (r)  =  (r  -  A)  (r  -  r3)  ...  (r  -  rn) 

=  c0  +  ctr  +  ...  c^^"-1 

and  multiplying  the  equations  by  c0,  clt  ...  cn_t  respectively  we  find  that, 
since  Cn_r  =  1, 

G1  (r,  -\)F  (r,)  +  Cz  [F  (r,)  +  (r,  -  A)  F'  (r,)]  =  af  (r). 

The  quantity  A  is  at  our  disposal.   Let  us  first  write  A  =  r1}  we  then 

have  /-i  n  /    \        f  i  •, 

CtF  (rj  =  af(r). 

Simplifying  the  preceding  equation  with  the  aid  of  this  relation  we 

°btain  C,F  (r,)  +  C2F'  (r,)  =  0. 

Writing  G  (r)  =  ajF  (r), 

we  have  C,  =  /  (T)  G'  (r,),     Ct  =f  (T)  O  (r,). 

These  are  just  the  constants  obtained  by  writing 


<f>(r)     r-fj      (r-rtf     r  -  r3      ">-rn' 

and  a  similar  rule  holds  in  the  case  of  a  multiple  root  of  any  order  or  any 
number  of  multiple  roots.  Thus  in  the  case  of  a  triple  root, 


§  1-49.   Heaviside's  expansion.  The  system  of  differential  equations  (II) 
of  §  1-47  may  be  written  in  the  form 


2  af.C.=X(0, 


where  the  a's  are  analogous  to  the  operators  ~rs  of  §  1-45. 


60  The  Classical  Equations 

Denoting  the  determinant  |  ar<s  \  by  <f>  (D)  and  using  Ars  to  denote  the 
co-factor  of  the  constituent  ars  in  this  determinant,  we  have 

<£  (D)  Q,  =   2  ArsEr  (t). 

r=l 

To  obtain  an  expansion  for  Qs  we  first  solve  the  equation 

j  (D)  ys  (t)  =  1 
with  the  supplementary  conditions 

y,(0)  =  y/(0)  =  ...y.<«»-»(0)=o, 
then  xrs  =  Asry3  (t) 

is  a  particular  solution  of  the  equation  </>  (D)  xr  —  Asr  .  1  for  which 

a:r(0)  =  *r'(0)  =  0; 

and  by  the  expansion  theorem  of  §  1-48, 
-  xrs  (t,  0)  =  Asrys  (t,  0) 

=    s    A_.(!-Ler..+  ^-0> 
jT,(ra)^(ra)e     +  .£(0)  ' 

which  is  Heaviside's  expansion  formula.  The  corresponding  formula  for 

<UOi- 

&  (0  =    S       #r  (0)  *ir  (*,  0)  +       tf/  (r)  *sr  (t,  r)  dr    , 

r-l    L  JO  J 

and  this  particular  solution  satisfies  the  conditions 

Qs  (0)  =  Q.'  (0)  =  o. 

§  1-51.  The  simple  wave-equation.  There  are  a  few  partial  differential 
equations  which  occur  so  frequently  in  physical  problems  that  they  may 
be  called  classical.  The  first  of  these  is  the  simple  wave-equation 


which  occurs  in  the  theory  of  a  vibrating  string  and  also  in  the  theory  of 
the  propagation  of  plane  waves  which  travel  without  change  of  form. 
These  waves  may  be  waves  of  sound,  elastic  waves  of  various  kinds,  waves 
of  light,  electromagnetic  waves  and  waves  on  the  surface  of  water.  In 
each  case  the  constant  c  represents  the  velocity  of  propagation  of  a  phase 
of  a  disturbance.  The  meaning  of  phase  may  be  made  clear  by  considering 
the  particular  solution 

V  =  sin  (x  —  ct) 

which  shows  that  F  has  a  constant  value  whenever  the  angle  x  —  ct  has 
a  constant  value.  This  angle  may  be  called  the  phase  angle,  it  is  constant 
for  a  moving  point  whose  x-co-ordinate  is  given  by  an  equation  of  type 
x  =  ct  -I-  a,  where  a  is  a  constant.  This  point  moves  in  one  direction  with 
uniform  velocity  c.  There  is  also  a  second  particular  solution 

V  =  sin  (x  -\-  ct) 


Wave  Propagation  61 

for  which  the  phase  angle  x  -f  ct  is  constant  for  a  point  which  moves  with 
velocity  c  in  the  direction  for  which  x  decreases.  These  solutions  may  be 
generalised  by  multiplying  the  argument  x  ±  ct  by  a  frequency  factor 
27TV/C,  where  v  is  a  constant  called  the  frequency,  by  adding  a  constant  y 
to  the  new  phase  angle  and  by  multiplying  the  sine  by  a  factor  A  to 
represent  the  amplitude  of  a  travelling  disturbance.  In  this  way  the 
particular  solution  is  made  more  useful  from  a  physical  standpoint  be- 
cause it  involves  more  quantities  which  may  be  physically  measurable.  In 
some  cases  these  quantities  may  be  more  or  less  determined  by  the  supple- 
mentary conditions  which  go  with  the  equation  when  it  is  derived  from 
physical  principles  or  hypotheses. 

Usually  this  particular  equation  is  derived  by  the  elimination  of  the 
quantity  U  from  two  equations 

dv      du     zu  _    dv  ... 

~di~adx'    ~dt~pfa  ......  IA) 

involving  the  quantities  U  and  F,  the  coefficients  a  and  ft  being  constants. 
The  constant  c  is  now  given  by  the  equation 

c2  -  aft. 

It  should  be  noticed  that  if  F  is  eliminated  instead  of  17,  the  equation 
obtained  for  U  is 


and  is  of  the  same  type  as  that  obtained  for  V.  This  seems  to  be  a  general 
rule  when  the  original  equations  are  linear  homogeneous  equations  of  the 
first  order  with  constant  coefficients,  however  many  equations  there  may 
be.  The  rule  breaks  down,  however,  when  the  coefficients  are  functions  of 
the  independent  variables.  If,  for  instance,  a  and  ft  are  functions  of  x  the 
resulting  equations  are  respectively 

327      a 


cw    .  a  /  du\ 

dt2  dx  \    dx  /  ' 

These  equations  may  be  called  associated  equations.  Partial  differential 
equations  of  this  type  occur  in  many  physical  problems.  If,  for  instance, 
y  denotes  the  horizontal  deflection  of  a  hanging  chain  which  is  performing 
small  oscillations  in  a  transverse  direction,  the  equation  of  vibration  is 

82y  __    a  /  dy\ 


where  g  is  the  acceleration  of  gravity  and  x  is  the  vertical  distance  above 
the  free  end.  Equations  of  the  above  type  occur  also  in  the  theory  of  the 
propagation  of  shearing  waves  in  a  medium  stratified  in  horizontal  plane 
layers,  the  physical  properties  of  the  medium  varying  with  the  depth. 


62  The  Classical  Equations 

§  1-52.  The  differential  equation  (I)  was  solved  by  d'Alembert  who 
showed  that  the  solution  can  be  expressed  in  the  form 

V=f(x-ct)  +  g(x  +  ct), 

where  /  (z)  and  g  (z)  are  arbitrary  functions  of  z  with  second  derivatives 
/"  (z)>  $"  (z)  that  are  continuous  for  some  range  of  the  real  variable  z. 
A  solution  of  type  f  (x  —  ct)  will  be  called  a  "  primary  solution,"  a  term 
which  will  be  extended  in  §  1-92  to  certain  other  partial  differential 
equations. 

To  illustrate  the  way  in  which  primary  solutions  can  be  used  to  solve 
a  physical  problem  we  consider  the  transverse  vibrations  of  a  fine  string 
or  the  shearing  vibration  of  a  building*. 

The  co-ordinate  x  is  supposed  to  be  in  the  direction  of  the  undisturbed 
string  and  in  the  vertical  direction  for  the  building,  the  co-ordinate  y  is 
taken  to  represent  the  transverse  displacement.  If  A  denotes  the  area  of 
the  cross-section,  which  is  a  horizontal  section  in  the  case  of  the  building, 
and  p  the  density  of  the  material,  the  momentum  of  the  slice  Adx  is  M  dx, 

where  M  —  pA  ^  .  The  slice  is  acted  upon  by  two  shearing  forces  acting 

in  a  transverse  direction  and  by  other  forces  acting  in  a  "vertical" 
direction,  i.e.  in  the  diiection  of  the  undisturbed  string.  Denoting  the 

shearing  force  on  the  section  x  by  S,  that  on  the  section  x  -f  dx  is  S  -f  ~    dx. 

dS  dx 

The  difference  is  x    dx,  and  so  the  equation  of  motion  is 
ox 

m  __  ss 

St   ~dx' 

We  now  adopt  the  hypothesis  that  when  the  displacement  y  is  very 
small  ~ 


where  /JL  is  a  constant  which  represents  the  rigidity  of  the  material  in  the 
case  of  the  building  and  the  tension  in  the  case  of  the  string.  According 
to  this  hypothesis  if  p  and  A  are  also  constants 


where  c2  =  p/p.  The  expressions  for  M  and  S  also  give  the  equation 

as      BM 
p-ft-P-to' 

and  so  we  have  two  equations  of  the  first  order  connecting  the  quantities 
M  and  8',  these  equations  imply  that  M  and  S  satisfy  the  same  partial 
differential  equation  as  y. 

i 

*  The  shearing  vibrations  of  a  building  have  been  discussed  by  K.  Suyehiro,  Journal  of  the 
Institute  of  Japanese  Architects,  July  (1926). 


Transmission  of  Vibrations  63 

In  the  case  of  the  building  one  of  the  boundary  conditions  is  that  there 
is  no  shearing  force  at  the  top  of  the  building,  therefore  8  =  0  when  x  =  h. 
Assuming  that 

y=f(x-ct)  +  g(x  +  ct) 

di/ 
the  condition  is  ^    =  0  when  x  =  h.  and  so 

ox 

0  =  /'  (A  -  ct)  +  g'  (h  +  ct). 
This  condition  may  be  satisfied  by  writing 

y  =  <f>  (ct  +  x  —  h)  ~h  (f>  (ct  —  x  +  7^), 

where  </>  (2)  is  an  arbitrary  function. 

A  motion  of  the  ground  (x  =  0)  which  will  give  rise  to  a  motion  of  this 
kind  is  obtained  by  putting  x  =  0  in  the  above  equation. 

Denoting  the  motion  of  the  ground  by  y  =  F  (t)  we  have  the  equation 

F  (t)  =  <f>  (ct  -  h)  -\-  </>  (ct  4-  h)  ......  (A) 

for  the  determination  of  the  function  </>  (z). 

In  the  case  of  the  string  the  end  x  =  /  may  be  stationary.  We  therefore 
put  y  =  0  for  x  =  I  and  obtain  the  equation 

0=f(l-ct)  +  g(l  +  ct) 
which  is  satisfied  by 


where  $  is  another  arbitrary  function.    If  the  motion  of  the  end  x  =  0  is 
prescribed  and  is  y  =  0  (t)  we  have  the  equation 

-0  (t)  =  </r  (Ct  -  1)  -  $  (Ct  +  I)  ......  (B) 

for  the  determination  of  the  function  $  (z). 

If,  on  the  other  hand,  the  initial  displacement  and  velocity  are  pre- 
scribed, say  ~ 

y=e(x),     gf  =  *(«) 

when  t  =  0,  we*have  the  equations 

0(x)=f  (x)  +  g  (x),     x  (x)  =  c  [<?'  (x)  -  /'  (x)] 
which  give  2cf  (x)  =  c9'  (x)  -  x(x)> 

2cg'  (x)  =  cff  (x)  +  x  (x), 
and  the  solution  takes  the  form 

1     rx+ct 

y=\\0(x~ct)  +  e(x  +  ct)-\+   f\       x(T)dr. 

L(j  J  x  -  ct 

If  in  the  preceding  case  both  ends  of  the  string  are  fixed,  the  equation 
(B)  implies  that  ^  (x)  is  a  periodic  function  of  period  21,  the  corresponding 
time  interval  being  2l/c.  Submultiples  of  these  periods  are,  of  course, 
admissible,  and  the  inference  is  that  a  string  with  its  ends  fixed  can  perform 
oscillations  in  which  any  state  of  the  system  is  repeated  after  every  time 


64  The  Classical  Equations 

interval  of  length  2lm/nc,  where  m  and  n  are  integers,  n  being  a  constant 
for  this  type  of  oscillation. 

In  the  case  of  the  building  the  ground  can  remain  fixed  in  cases  when 
(f>  (z)  is  a  periodic  function  of  period  4=h  such  that 

<f)  (z  -f  2h)  =  —  <f)  (z). 

It  should  be  noticed  that  the  conditions  of  periodicity  may  be  satisfied 
by  writing 

0  (z)  =  sin  -  j-  ,     (f>  (z)  =  sin  [(n  +  |)  TTZ/&], 

where  in  and  n  are  integers.  Thus  in  the  case  of  the  string  with  fixed  ends 
there  are  possible  vibrations  of  type 


mirx       irnrct 
-       cos 


y-~, 

and  in  the  case  of  the  building  with  a  free  top  and  fixed  base  there  are 
possible  vibrations  of  type 

.    .  r/     iv7^"!     r/     ivfrttfi 

y  =  bm  sin  \(n  +  J)  ,    |  cos  \(n  -f  J)  -,-    . 

L  ^  J         L  ^  J 

These  motions  may  be  generalised  by  writing  for  the  case  of  the  string 

«         .    mnx       mnct 
y  -  S  am  sin       -  cos  —  y—  , 

m-l  *  * 

where  the  coefficients  aw  are  arbitrary  constants.  For  complete  generality 
we  must  make  s  infinite,  but  for  the  present  we  shall  treat  it  as  a  finite 
constant.  The  total  kinetic  energy  of  the  string  is 


,  [i   .    mnx   .    HTTX  -, 

since  we  have  sm     7    sm  — .    ax  =  0          n  +  m 

Jo          *  * 

=  Z/2        ft  =  m. 

Since  the  kinetic  energy  is  the  sum  of  the  kinetic  energies  of  the  motions 
corresponding  to  the  individual  terms  of  the  series,  these  terms  are  supposed 
to  represent  independent  natural  vibrations  of  the  string.  These  are  generally 
called  the  normal  vibrations. 

The  solution  for  the  vibrating  building  can  also  be  generalised  so  as  to 
give  3 

y=  S  bn  sin  \(n  -f  J)  \-    cos  \  (n  -f  i)  -,-  L  • 
n-i  L  ^J         L  h  \ 

and  the  kinetic  energy  is  in  this  case 

f,     (271-f    1)27T2C2,     2      . 

>          ^  '  *x      2    nt  v^ 


Vibration  of  a  String  65 

for  now  we  have  corresponding  relations 

/    sin  \(n  4-  J)  ^-    sin    (m  -f  J)  ^-  ute  =  0  m±n 

=  A/2        m  —  n. 
Such  relations  are  called  orthogonal  relations. 

§  1-53.   In  the  case  of  the  equation  of  the  transverse  vibrations  of  a 
string  there  is  a  type  of  solution  which  can  be  regarded  as  fundamental. 
Let  us  suppose  that  the  point  x  =  a  is  compelled  to  move  with  a  simple 

harmonic  motion*  n        ,   t        . 

y  =  p  cos  (pt  4-  «), 

where  a,  /3  and  p  are  arbitrary  real  constants.    If  the  ends  x  =  0,  x  =  I 
remain  fixed,  it  is  easily  seen  that  the  differential  equation 


W          dx*  ' 

and  the  conditions  y  =  0  at  the  ends  may  be  satisfied  by  writing  y  —  y±  for 
0  <  x  <  a  and  y  =  y2  for  a  <  x  <  /,  where 

yl  =  p  cosec  (Xa)  sin  (\x)  cos  (p£  -fa), 

2/2  =  /?  cosec  A  (/  —  a)  sin  A  (/  —  #)  cos  (pt  +  a), 

and  Ac  =  p.  The  case  of  a  periodic  force  F  =  F0  cos  (^tf  -f  a)  concentrated 
on  an  infinitely  short  length  of  the  string  may  be  deduced  by  writing  down 
the  condition  that  the  forces  on  the  element  must  balance,  the  inertia 
being  negligible.  This  condition  is 

F  =  Py/  -  PyJ        for  x  =  a, 
where  P  is  the  pull  of  the  string.   Substituting  the  values  of  y^  and  y%  we 

get  cF0  =  pfiP  [cot  Aa  +  cot  A  (I  -  a)]  . 

F 
Therefore  /?  =  p^  cosec  XI  sin  Aa  sin  A  (I  —  a). 

The  solution  can  now  be  written  in  the  form 

F 

y=  j>0  (*>«)> 

,  ,       .      sin  A#  sin  A  (Z  —  a) 

where  <7  (x,  a)  =  -  c—  ^—  ^  --  -        0  <  x  <  a 

A  sin  AL 

_  sin  A  (I  —  x)  sin  Aa  7  ' 

=  -----  x-  —  .  —  ^j  -----         a  ^  x  ^  /. 
A  sin  XI 

This  function  gr  (a;,  a)  is  a  solution  of  the  differei^tial  equation 

S+*V-0  ......  (A) 

*  Rayleigh,  Theory  of  Sound,  vol.  I,  p.  195 


66  The  Classical  Equations 

and  satisfies  the  boundary  conditions  g  =  0  when  x  —  0  and  when  x  —  I. 
It  is  continuous  throughout  the  interval  0  <  x  <  I,  but  its  first  derivative 
is  discontinuous  at  the  point  x  =  a  and  indeed  in  such  a  manner  that 

lim 

8->Q 

The  function  g  (x,  a)  is  called  a  Green's  function  for  the  differential 

d2u 
expression    ,  ^  4-  A2w,  it  possesses  the  remarkable  property  of  symmetry 

expressed  by  the  relation 

g  (x,  a)  =  g  (a,  x). 

This  is  a  particular  case  of  the  general  reciprocal  theorem  proved  by 
Maxwell  and  the  late  Lord  Rayleigh. 

It  should  be  noticed  that  the  Green's  function  does  not  exist  when  A 
has  a  value  for  which  sin  XI  =  0?  that  is,  a  value  for  which  the  equation  (A) 
possesses  a  solution  g  =  sin  Xx  which  satisfies  the  boundary  conditions  and 
is  continuous  (D,  1)  throughout  the  range  (0  <  x  <  /). 

A  fundamental  property  of  the  Green's  function  g  (x,  a)  is  obtained  by 
solving  the  differential  equation 


by  the  method  of  integrating  factors.  Assuming  that  y  is  continuous  (Z>,  2) 
in  the  interval  (0,  1)  and  that  the  function/  (x)  is  continuous  in  this  interval, 
the  result  is  that 


U  =  U    °  -  U 

a-0 


~ 

l      Jo 


where  u  (0)  and  u  (I)  are  assigned  values  of  u  at  the  ends.  If  these  values 
are  both  zero  y  is  expressed  simply  as  a  definite  integral  involving  the 
Green's  function  and/  (a). 

§  1-54.  The  torsional  oscillations  of  a  circular  rod  are  very  similar  in 
character  to  the  shearing  oscillations  of  a  building.  Let  us  consider  a 
straight  rod  of  uniform  cross-section,  the  centroids  of  the  sections  by 
planes  x  =  constant,  perpendicular  to  the  length  of  the  rod,  being  on  a 
straight  line  which  we  take  as  axis  of  x.  Let  us  assume  that  the  section  at 
distance  x  from  the  origin  is  twisted  through  an  angle  6  relative  to  the 
section  at  the  origin.  It  is  on  account  of  the  variation  of  8  with  x  that 
an  element  of  the  rod  must  be  regarded  as  strained.  The  twist  per  unit 
length  at  the  place  x  is  defined  to  be 

B8 

T  =  ^; 
it  vanishes  when  8  is  constant  throughout  the  element  bounded  by  the 


Torsional  Oscillations  67 

planes  x  and  x  -f  dx,  i.e.  when  this  element  is  simply  in  a  displaced  position 
just  as  if  it  had  been  rotated  like  a  rigid  body. 

The  torque  which  is  transmitted  from  element  to  element  across  the 
plane  x  is  assumed  to  be  Kfir,  where  /A  is  an  elastic  constant  for  the  material 
(the  modulus  of  rigidity)  and  K  is  a  quantity  which  depends  upon  the  size 
and  shape  of  the  cross-section  and  has  the  same  dimensions  as  7,  the 
moment  of  inertia  of  the  area  about  the  axis  of  x. 

Let  p  denote  the  density  of  the  material,  then  the  moment  of  inertia 
abou^  the  axis  of  x  of  the  element  previously  considered  is  pldx  and  the 
angular  momentum  is  plOdx. 

Equating  the  rate  of  change  of  angular  momentum  to  the  difference 
between  the  torques  transmitted  across  the  plane  faces  of  the  element,  we 
obtain  the  equation  of  motion 

.  d26         „  cW 
pi        =  fih  „ 
cl2       ^     fix* 

which  holds  in  the  case  when  the  rod  is  entirely  free  or  is  acted  upon  by 
forces  and  couples  at  its  ends.  In  this  case  the  differential  equation  must 
be  combined  with  suitable  end  conditions. 

A  simple  case  of  some  interest  is  that  in  which  the  end  x  —  0  is  tightly 
clamped,  whilst  the  motion  of  the  other  end  x  —  a  is  prescribed. 

§  1-55.  The  same  differential  equation  occurs  alstfin  the  theory  of  the 
longitudinal  vibrations  of  a  bar  or  of  a  mass  of  gas. 

Consider  first  the  case  of  a  bar  or  prism  whose  generators  are  parallel 
to  the  axis  of  x.  Let  x  -f  £  denote  the  position  at  time  t  of  that  cross- 
section  whose  undisturbed  position  is  x,  then  £  denotes  the  displacement 
of  this  cross-section.  An  element  of  length,  8x,  is  then  altered  to  8  (x  -f-  f ), 
of  (1  -f  £')  $x,  where  the  prime  denotes  differentiation  with  respect  to  x. 
Equating  this  to  (1  +  e)  $x  we  shall  call  e  the  strain.  The  strain  is  thus  the 
ratio  of  the  change  in  length  to  the  original  length  of  the  element  and  is 
given  by  the  formula  ^ 

e  =  ' 

dx 

According  to  Hooke's  law  stress  is  proportional  to  strain  for  small 
displacements  and  strains.  The  total  force  acting  across  the  sectional  area 
A  in  a  longitudinal  direction  is  therefore  F  =  EeA,  where  E  is  Young's 
modulus  of  elasticity  for  the  material  of  which  the  rod  is  composed.  The 
stress  across  the  area  is  simply  Ee. 

The  momentum  of  the  portion  included  between  the  two  sections  with 

co-ordinates  x  and  x  -f  8x  is  M  8x,  where  M  =  pA  ~  and  p  is  the  density  of 
the  material.  The  equation  of  motion  is  then 

<W     W 
dt  ~  dx9 

5-2 


68  The  Classical  Equations 

AS*€ 

A' 


When  the  material  is  homogeneous  and  the  rod  is  of  uniform  section  the 
equation  is  ^ 

a^^cte5' 

where  c2  =  £//o.  Since  the  modulus  E  for  most  materials  is  about  two  or 
three  times  the  modulus  of  rigidity  /z,  longitudinal  waves  travel  much  more 
rapidly  than  shearing  waves  and  the  frequency  of  the  fundamental  mode 
of  vibration  is  higher  for  longitudinal  oscillations  than  it  is  for  shearing 
oscillations.  In  the  case  of  a  thin  rod  shearing  oscillations  would  not  occur 
alone  but  would  be  combined  with  bending,  and  the  motion  is  different. 

The  fundamental  frequency  for  the  lateral  oscillations  is,  however,  much 
lower  than  that  for  the  longitudinal  oscillations.  Let  us  next  consider  the 
propagation  of  plane  waves  of  sound  in  a  direction  parallel  to  the  axis  of  x. 

Let  VQ  =  A  8x  be  the  initial  volume  of  a  disc-shaped  mass  of  the  gas 
through  which  the  sound  travels,  v  =  A8  (x  -f  £)  the  volume  of  the  same 
mass  at  time  t.  We  then  have 

v  =  vQ(l  -f  e), 

where  e  is  now  the  dilatation.  If  pQ  is  the  original  density  and  p  the  density 
of  the  mass  at  time  t,  we  may  write 

P  =  PQ  (1  +  s), 

where  s  is  the  condensation,  it  is  the  ratio  of  the  increment  of  density  to 
the  original  density.  Since  pv  =  pQvQ  we  have 

(1  +  s)  (1  +  e)  =  1, 
and  if  s  and  e  are  both  small  we  may  write 


~  . 
ox 

To  obtain  the  equation  of  motion  we  assume  that  the  pressure  varies 
with  the  density  according  to  some  definite  law  such  as  the  adiabatic  law 


where  p0  is  the  pressure  corresponding  to  the  density  y  and  is  a  constant 
which  is  different  for  different  gases. 

This  law  holds  when  there  is  no  sensible  transfer  of  heat  between 
adjacent  portions  of  the  gas.  Such  a  state  of  affairs  corresponds  closely 
to  the  facts,  since  in  the  case  of  vibration  of  audible  frequency  the  con- 
densations and  rarefactions  of  our  disc-shaped  mass  of  gas  follow  one 
another  with  a  frequency  of  500  or  more  per  second. 

For  small  values  of  s  we  may  write 

P  =  Po(l  +  7s)- 


Waves  of  Sound  69 

The  equation  of  motion  is  now 

m  _SF 

St  ~  Sx  ' 
where  M  =  PoA  ^,    F  =  -  Ap. 

Substituting  the  values  of  p  and  s  we  obtain  the  equation 

a«f_    a«f 

~ 


in  which  c»  =  y° 

Po 

For  sound  waves  in  a  tube  closed  at  both  ends  the  boundary  conditions 
are  £  =.  0  when  x  =  0  and  when  #  =  I.  The  solution  is  just  the  same  as  the 
solution  of  the  problem  of  transverse  vibration  of  a  string  with  fixed  ends. 

For  sound  waves  in  a  pipe  open  at  both  ends  and  for  the  longitudinal 
vibrations  of  a  bar  free  at  both  ends  we  have  the  boundary  conditions 


when  x  =  0  and  when  x  =  19  which  express  that  there  is  no  stress  at  the 
ends.  The  normal  modes  of  vibration  are  now  of  type 

..      ~          nnrx        (rmrct\ 

g=Cm  COS  -j-    COS  ^-y-  J  , 

where  Cm  is  an  arbitrary  constant  and  m  is  an  integer.  This  solution  is  of 
type  £  =  <l>(x  +  ct)  +  <f>(ct-  x), 

and  may  be  interpreted  to  mean  that  the  progressive  waves  represented 
by  ^  =  <f>  (ct  —  x)  are  reflected  at  the  end  x  =  0  with  the  result  that  there 
is  a  superposed  wave  represented  by  £>  =  <£  (ct  -f  x). 

There  is  a  different  type  of  reflection  at  a  closed  end  of  a  tube  (or  fixed 
end  of  a  rod),  as  may  be  seen  from  the  solution 

£  =  ^  (ct  -  x)  -  <f>  (ct  +  x), 

which  makes  f  =  0  when  x  =  0. 

Reflection  at  a  boundary  between  two  different  fluid  media  or  between 
two  parts  of  a  bar  composed  of  different  materials  may  be  treated  by 
introducing  the  boundary  condition  that  the  stress  and  the  velocity  must 
be  continuous  at  the  boundary. 

If  progressive  waves  represented  by  £0  =  #0</>  (t  —  x/c)  approach  the 
boundary  x  =  0  from  the  negative  side  and  give  rise  to  a  reflected  wave 
£x  =  a^  (t  4-  x/c)  and  a  transmitted  wave  £2  =  ^2^  (^  "~  #/c/)>  the  boundary 
conditions-  are  fit  fit  fit 

' 


"  ** 


** 


70  The  Classical  Equations 

where  K  =  yp0  and  K  =  y'p0  ,  the  constants  y  and  y'  referring  respectively 
to  the  media  on  the  negative  and  positive  sides  of  the  origin.  The  equi- 
librium pressure  pQ  is  the  same  for  both  media. 

9&  9fi  9& 

Now  -*  =  «0,     -g^  =-<»i,      *  =  c'*, 

hence,  when  x  =  0, 

c$0  -  o^'  (0,     -  cs!  =  <*!</>'  (t),     c'3ij=  a^  (t), 
and  c  (s0  —  s^)  =  c's2,     K  (SQ  -\-  Sj)  —  i<'s. 


mi  .  — 

Iherefore  sl       —  —      «s0       $  =  -  —        5 

1        jc'c-f  KC'    °  2       //C-f  ice'    °' 


—  KC 


t  ,  ,  /  • 

KC+  KC'  /c'C+  ACC7 

§  1-56.  The  simple  wave-equation  occurs  also  in  an  approximate  theory 
of  long  waves  travelling  along  a  straight  canal,  with  horizontal  bed  and 
parallel  vertical  sides,  the  axis  of  x  being  parallel  to  the  vertical  sides  and 
in  the  bed  (see  Lamb's  Hydrodynamics,  Oh.  vm). 

Let  6  be  the  breadth  of  the  canal  and  h  the  depth  of  the  fluid  in  an 
initial  state  at  time  t  —  0  when  the  fluid  is  at  rest  and  its  surface  horizontal. 
We  shall  denote  the  density  of  the  fluid  by  p  and  the  pressure  at  a  point 
(x,  y,  z)  by  p.  The  motion  is  investigated  on  the  assumption  that  p  is 
approximately  the  same  as  the  hydrostatic  pressure  due  to  the  depth 
below  the  free  surface.  This  means  that  we  write 

P  =  Po  +  9P  (h  +  ri-  y),  ......  (I) 

where  ^pQ  is  the  external  pressure,  which  is  supposed  to  be  uniform,  77  is 
the  elevation  of  the  free  surface  above  its  undisturbed  position  and  g  is 
the  acceleration  of  gravity.  One  consequence  of  this  assumption  is  that 
there  is  no  vertical  acceleration,  in  other  words,  the  vertical  acceleration  is 
neglected  in  making  this  approximation. 

If,  in  fact,  we  consider  a  small  element  of  fluid  bounded  by  horizontal 
and  vertical  planes  parallel  to  the  planes  of  reference,  the  axis  of  y  being 
vertically  upwards,  the  equations  of  motion  are 

pa  .  Sx8ySz  =  —  ~    8x  .  8y$z, 

pft  .  8x8ySz  =  —  ~  -  Sy  .  8z8x  —  pg8x8y8z, 

d%) 
py  .  8x8y8z  =  —  «    8z  .  8x8y, 

where  a,  j8,  y  are  the  component  accelerations.  With  the  above  assumption 

we  have  8  =  y  =  0,  and  so  ^ 

op 


Waves  in  a  Canal 


71 


The  assumption  of  no  vertical  acceleration  is  not  equivalent  to  the 
assumption  (I),  because  an  arbitrary  function  of  x,  z  and  t  could  be  added 
to  the  right-hand  side  of  (I)  and  the  equations  of  motion  would  still  give 
no  vertical  acceleration. 

Equation  (I)  gives  pa  =  —  gp  x-  . 

This  expression  for  2  is  independent  of  y,  consequently,  since  g  is 
assumed  to  be  constant,  the  acceleration  a  is  the  same  for  all  particles  in 
a  vertical  plane  perpendicular  to  the  axis  of  x.  The  horizontal  velocity  u 
depends  on  x  and  t  only. 

Now  let  £  be  the  total  displacement  from  their  initial  position  of  the 
particles  which  at  time  t  occupy  the  vertical  plane  x.  Each  particle  is 
supposed  to  have  moved  horizontally  through  a  distance  £,  but  actually 
some  of  the  particles  will  have  moved  slightly  upwards  or  downwards  as  well. 

the  fluid  which  occupies  the  region  QQ'X'X  is 
supposed  to  have  initially  occupied  the  region 
PP'A'A. 

Equating  the  amount  of  fluid  in  the  region 
QQ'N'N  to  the  difference  of  the  amounts  in  the 
regions  PNXA,  P'N'X'A'  we  obtain  the  equa- 
tion of  continuity 


N 


'N' 


A  A  X  X' 

Fig.  9. 


or 


i 

=  -  h  —- . 

dx 


.(II) 


A  second  equation  is  obtained  by  writing  a  =  >,— .  This  is  approximately 

ot 

true  in  the  case  of  infinitely  small  motions,  the  exact  equation  being 


a  — 


du         Bu 


u 


dt^"dx 

?=!*&, 

du 


Writing 

we  have  |^2  =  ~~  =  a  =  -  g  g.  (Ill) 

The  equations  (II)  and  (III)  now  give  the  wave-equations 

where  c2  =  gh. 

When,  in  addition  to  gravity,  the  fluid  is  acted  upon  by  small  dis- 
turbing forces  with  components  (X,  Y)  per  unit  mass  of  the  fluid,  the 


72  The  Classical  Equations 

assumption  that  the  pressure  is  approximately  equal  to  the  hydrostatic 
pressure  leads  to  the  equation 

- 

(g-Y)dy. 


v 


TU-      •  P        i        vx*7 

Thisgives  £-/>&-  Y^x 

and  the  equation  of  horizontal  motion 

du         „      dp 

'•-fc-^-fc 

indicates  that  in  general  u  depends  on  y  as  well  as  on  x  and  t. 

With,  however,  the  simplifying  assumptions  that  Y  is  small  compared 

dY 
with  g  and  that  h  ~     is  small  in  comparison  with  X  the  equation  takes  the 

form  3  _ 


and,  if  X  depends  only  on  x  and  t,  this  equation  indicates  that  u  is  inde- 
pendent of  y.   We  may  then  proceed  as  before  and  obtain  the  equations 


EXAMPLES 

1.   An  elastic  bar  of  length  I  has  masses  m0,  m1  at  the  ends  x  —  0,  x  —  I  respectively. 
Prove  that  the  terminal  conditions  are 

Jj]  A    -  _  —  vn  wVi  (*T\    y  ' — .  0 

^^  o-  ~  mQ  1*9.         wiit-ii    &  —  v, 


Prove  that  the  possible  frequencies  of  vibration  are  given  by  the  equation 

(1  -  KHiP)  tan  9  +  U  +  /*i)  ^  =  0, 

where  c2m^  =  lAE^t     c2ml  =  lAE^ilt     0  =  nl, 

and  nc/27T  is  the  number  of  vibrations  per  second. 

2.  If  a  prescribed  vibration  £  =  C  cos  w£  is  maintained  at  the  end  x  =  0  of  a  straight 
pipe  which  is  closed  at  the  end  x  =  I  the  vibration  at  the  place  x  is  given  by 

~  nl  .    n  (I  —  x) 

f  =  C  cosec  -  sin  — cos  nt. 

c  c 

Obtain  the  corresponding  solution  for  the  case  in  which  the  end  x  =  I  is  open. 

3.  Discuss  the  longitudinal  oscillations  of  a  weighted  bar  whose  upper  end  is  fixed. 


Systems  of  Equations  73 

4.  If       ^ 


and  A  is  an  arbitrary  constant,  the  function 

y  —  A  [sin  2sna  —  sin 

satisfies  the  differential  equation  _0  _0 

d*y  _      d*y 

2  ~          z  ' 


and  the  end  conditions  y  —  0  when  x  =  0  and  when  x  =  a  +  vt.  Prove  also  that  when  v  ->  0, 


0  ,     .     SnX         Snct 

it  ->  2  A  sm  —  cos  —  . 
a  a 


[T.  H.  Havelock,  Phil.  Mag.  vol.  XLVH,  p.  754  (1924).] 
5.   Prove  that  if  y  —  0  when  x  =  0  and  x  =  vt, 
y  =/(*).    y  =  9  (*)> 
a  solution  of    ~  ^  =  c2  ^~   is  given  by 


~  ^ 


where  a  log  --  =  2n, 

exp  (r)  =  ~  -  -£-      ^K  +  «)  =  i/(«)  -f  I    fX  g(x)dx, 

foz        Cz«o  ^c  .'  0 


Bn  -  -  -  f  *   f  K  +  *)  cos  (naa>)  -^-  -  , 

7T    J  -Vt0  MO  +  3? 

and  it  is  supposed  that         .  .       .  .  ,  % 

FF^  /(-*)  =  -/(*),    g(-*)--^(«). 

[E.  L.  Nicolai,  PM.  Jfogr.  vol.  XLIX,  p.  171  (1925).] 


§  1-61.  Conjugate  functions  and  systems  of  partial  differential  equations. 
If  in  equations  ((A)  §  1-51)  we  write  a  =  1,  /?  =  -  1  and  use  the  variable 
y  in  place  of  t  we  obtain  the  equations 

W^dV      dU^^dV 

dx       dyy      dy  ~       ~dx 

satisfied  by  two  conjugate  functions  U  and  V.  In  this  case  both  functions 
satisfy  the  two-dimensional  form  of  Laplace's  equation 


_ 

dx*      3^2  ~    - 

This  equation  is  important  in  hydrodynamics  and  in  electricity  and 
magnetism. 

The  equations  (A)  may  be  generalised  in  another  way  by  writing 

.97        dU        3V 


74  The  Classical  Equations 

where  a,  /?,  y,  8,  6,  </>,  A,  /x,  a,  r  are  arbitrary  constants.    In  particular, 

the  equations  w         ^V      dU 

dt  ==SK'dx'     ~dx^V 


i     A+    ^  f  dU 

lead  to  the  equation  -~--  =  K  ~  2  , 

which  is  the  equation  for  the  conduction  of  heat  in  one  direction  when  U 
is  interpreted  as  the  temperature  and  K  as  the  diffusivity.  The  same 
equation  occurs  in  the  theory  of  diffusion.  It  should  be  noticed  that  the 
quantity  V  satisfies  the  same  equation. 

Again,  if  we  write  -~—  =  L  -x-  -f  RU, 

du^cdv  +  sv 
dx     °  dt  ^     ' 

and  interpret  V  as  electric  potential,  U  as  electric  current,  we  obtain  the 
differential  equation 


which  governs  the  propagation  of  an  electric  current  in  a  cable*.  The 
coefficients  have  the  following  meanings  : 

R  L  C  -        S 

resistance  inductance  capacity  leakance 

all  per  unit  of  length  of  the  cable.  The  quantity  U  satisfies  the  same 

differential  equation  as  V.  This  differential  equation  may  be  reduced  to  a 

canonical  form  by  introducing  the  new  dependent  variables  u,  v,  defined 

by  the  equations  TT  mlT  T7  p./r 

J  ^  u  =  UeRtiL,     v  =  Vem<L. 

These  variables  satisfy  the  equations 

dv       ,-  du 


and  the  canonical  equations  of  propagation  are  Heaviside's  equations 


These  equations  are  of  the  simple  type  (I)  if 

SL  =  CR. 
In  this  case  a  wave  can  be  propagated  along  the  cable  without  distortion. 

*  Cf  .  J.  A.  Fleming,  The  Propagation  of  Electric  Currents  in  Telephone  and  Telegraph  Circuits,  ch.  v. 


The  Telegraphic  Equation  75 

When  dealing  with  the  general  equations  (A)  it  is  advantageous  to  use 
algebraic  symbols  for  the  differential  operators  and  to  write 

3     n      d  -D  • 
si    n"   te~Dx' 

the  differential  equations  may  then  be  written  symbolically  in  the  form 
(0Dt  -  yDx  -  /*)  F  =  (aD.  +  A)  U,     (<f>Dt  -Wx-a)U  =  (fiDx  +  r)  V. 
The  first  equation  may  be  satisfied  by  writing 

U=(ODt-yD.-rtW,     V=(aDx+\)W,        ......  (B) 

where  W  is  a  new  dependent  variable.  Substituting  in  the  second  equation 
we  obtain  the  following  equation  for  W, 

[(0Dt  -  yDx  -  p)  (^Dt  -  8DX  -  a)  ~  (aDx  +  A)  ($DX  +r)]W^  0, 

which,  when  written  in  full,  has  the  form 
9217  9217  921^ 

/v  /    f     rr  /  n&     ,       i      \  V     r"       .      /     &  /*>\  U     "  i  f\ 


-  (ar  -f  j3A  +  ycr  +  8/i)    ~  +  (AIO-  -  XT)  W  =  0. 

When  this  equation  has  been  solved  the  variables  U  and  V  may  be 
determined  with  the  aid  of  equations  (B).  It  is  easily  seen  that  U  and  V 
satisfy  the  same  equation  as  W. 

The  equation  for  W  is  said  to  be  hyperbolic,  parabolic  or  elliptic 
according  as  the  roots  of  the  quadratic  equation 

6</>X*  -  (68  +  fa)  X  +  yS  -  aft  -  0 

are  real  and  distinct,  equal  or  imaginary.  In  this  classification  the  co- 
efficients a,  /?,  y,  S,  By  fi,  A,  /z,  a,  r  are  supposed  to  be  all  real,  the  simple 
wave-equation  is  then  of  hyperbolic  type,  the  equation  of  the  conduction 
of  heat  of  parabolic  type  and  Laplace's  equation  of  elliptic  type.  The 
telegraphic  equation  is  generally  of  hyperbolic  type,  but  if  either  C  =  0  or 
L  =  0  it  is  of  parabolic  type  and  the  canonical  equation  is  of  the  same  form 
as  the  equation  of  the  conduction  of  heat. 

The  foregoing  analysis  requires  modification  if  the  coefficients  a,  j8,  y, 
8,  0,  <f>,  A,  /A,  a,  r  are  functions  of  x  and  t,  because  then  the  operators  aDx  -f-  A 
and  9Dt  —  yDx  —  \L  are  not  commutative  in  general,  and  so  the  first 
equation  cannot  usually  be  satisfied  by  means  of  the  substitution  (B).  If, 
however,  the  conditions  ^^ 


da 


76  The  Classical  Equations 

are  satisfied  the  operators  are  commutative  (permutable)  and  a  differential 
equation  may  be  obtained  for  W.  In  this  case  the  variables  U  and  F  do 
riot  necessarily  satisfy  the  same  partial  differential  equation.  This  is  easily 
seen  by  considering  the  simple  case  when  the  first  equation  is  U  =  0dV/dt 
and  /?  and  r  are  independent  of  t. 

Differential  operators  which  are  not  permutable  play  an  interesting 
part  in  the  new  mechanics. 

§  1-62.  For  some  purposes  it  is  useful  to  consider  the  partial  difference 
equations  which  are  analogous  to  partial  differential  equations  in  which 
we  are  interested.  The  notation  which  is  now  being  used  in  Germany  is 
the  following  *  : 

u  (x  +  h,  y)  -  u  (x,  y)  =  hux,     u  (x,  y  -f  h)  -  u  (x,  y)  =  huv, 
u  (x,  y}  —  u  (x  —  h,  y)  =  hu^,     u  (x,  y)  —  u  (x,  y  —  h)  =  hu^, 

u  (x  +  h,y)  —  2u  (x,  y)  -f  u  (x  —  h,  y)  =  Ji2ux^  =  h^u^x  - 
The  equations  ux  —  Vy>     uy  —  —  v% 

are  analogous  to  those  satisfied  by  conjugate  functions  since  they  imply 

Ux7   +   Vvy    =     0,  Vx*     +     Vyy     =      0. 

The  equations  %  —  vv  ,     u^  —  vx 

give  the  equations  ux2  =  u$  ,     vx^  =  vg 

analogous  to  the  equation  of  the  conduction  of  heat. 

§  1-63.  The  simultaneous  equations  from  which  the  final  partial 
differential  equation  is  derived  need  not  be  always  of  the  first  order.  In 
the  theory  of  the  transverse  vibrations  of  a  thin  rod  the  primary  equations 


where  77  is  the  lateral  displacement,^  the  bending  moment,  A  the  sectional 
area,  x  the  radius  of  gyration  of  the  area  of  the  cross-section  about  an  axis 
through  its  centre  of  gravity,  p  the  density  and  E  the  Young's  modulus 
of  the  material.  The  resulting  equation 


is  of  the  fourth  order.  The  equation  is  usually  simplified  by  the  omission 
of  the  second  term.  This  process  of  approximation  needs  to  be  carefully 
justified  because  it  will  be  noticed  that  the  term  omitted  involves  a 
derivative  of  the  fourth  order,  that  is  a  derivative  of  the  highest  order. 
Now  there  is  a  danger  in  omitting  terms  involving  derivatives  of  the  highest 

*  See  an  article  by  R.  Courant,  K.  Friedrichs  and  H.  Lewy,  Math.  Ann.  Bd.  c,  S.  32  (1928). 
f  Cf.  H.  Lamb,  Dynamical  Theory  of  Sound,  p.  121. 


Vibration  of  a  Rod  77 

order  because  their  coefficients  are  small.  This  may  be  illustrated  in  a  very 
simple  way  by  considering  the  equation 


where  v  is  small.  The  solution  is  of  type 

7]  =  A  -f  JBe*>, 

where  A  and  B  are  constants.  When  the  term  on  the  right  of  (IT)  is  omitted 
the  solution  is  simply  77  —  A.  When  x  and  v  are  both  small  and  positive 
the  term  Bexlv,  which  is  omitted  in  the  foregoing  method  of  approximation, 
may  be  really  the  dominant  term.  In  this  example  all  the  terms  involving 
derivatives  of  the  highest  order  have  been  omitted,  and  as  a  general  rule 
this  is  more  dangerous  than  the  omission  of  only  some  of  the  terms  as  in 
the  case  of  the  vibrating  rod.  The  omission  of  the  second  term  from  the 
rod  equation  seems  to  be  quite  justifiable  when  the  rod  is  very  thin.  When 
the  rod  is  thick  Timoshenko's  theory*  shows  that  there  is  a  term  giving 
the  correction  for  shear  which  is  at  least  as  important  as  the  second  term 
of  the  usual  equation  (I). 

This  point  relating  to  the  danger  of  omitting  terms  involving  derivatives 
of  the  highest  order  comes  up  again  in  hydrodynamics  when  the  question 
.of  the  omission  of  some  or  all  of  the  viscous  terms  comes  under  considera- 
tion. The  omission  of  all  the  viscous  terms  lowers  the  order  of  the  equations 
and  requires  a  modification  of  the  boundary  conditions.  This  does  riot  lead 
to  very  good  results.  On  the  other  hand,  in  PrandtFs  theory  of  the 
boundary  layer  some  of  the  viscous  terms  are  retained,  the  boundary 
condition  of  no  slipping  at  the  surface  of  a  solid  body  is  also  retained  and 
the  results  are  found  to  be  fairly  satisfactory. 

EXAMPLE 

r»  Au    j.  ^u  A-  &v  Bu        ,  du 

Prove  that  the  equations  -    =  a  -  -   -f  b  ~-  , 

ox        ox        oy 

dv  du  ,  du 
2  =  c  5-  -f  d  5- 
oy  ox  oy 

give  an  equation  of  the  second  order  which  is  elliptic,  parabolic  or  hyperbolic  according  as 
(a  —  d)2  +  46c  is  less  than,  equal  to  or  greater  than  zero. 

[E.  Picard,  CompL  Rend.  t.  cxn,  p.  685  (1891).] 

§  1-71.  Potentials  and  stream-functions.  The  classical  equations  are  of 
great  mathematical  interest  and  have  played  an  important  part  in  the 

*  Phil.  Mag.  (6),  vol.  XLI,  p.  744  (1921).  The  equation  used  by  Timoshenko  is  of  type 

„  23S?        d*T)         2 
EK  3~4  +  P  aTa  -  P" 

*     r    * 


where  jz  is  the  mocjultis  of  rigidity  and  a  is  a  constant  which  depends  upon  the  shape  of  the  cross  - 
section.    For  the  equation  of  resisted  vibrations  see  Note  II,  Appendix. 


78  The  Classical  Equations 

development  of  mathematical  analysis  by  suggesting  fruitful  lines  of 
investigation.  It  can  be  truly  said  that  the  modern  theory  of  functions 
owes  its  origin  largely  to  a  study  of  these  equations.  The  theory  of  functions 
of  a  complex  variable  is  associated,  for  instance,  with  the  theory  of  con- 
jugate functions  and  the  solutions  of  Laplace's  equation. 

If,  for  instance,  we  write 

0  +  ty=/(a;  +  ty)  =/(«), 

where/  (z)  is  an  analytic  function*  and  (f>  and  0  are  real  when  x  and  y  are 
real,  we  have,  for  points  in  the  domain  for  which/  (z)  is  analytic, 


= 
dy        dy      J 

where  f  (z)  denotes  the  derivative  of  /(z). 
These  equations  give 


Equating  the  real  and  imaginary  parts  of  the  two  sides  of  this  equation, 
we  see  that  ~  .      -  , 


d<f>          dJj 

V=  -  ar  =  ^,  say. 

oy  ox  J 

These  relations  between  the  derivatives  of  two  conjugate  functions  <f> 
and  ifj  are  called  Cauchy's  relations  because  they  play  a  fundamental  part 
in  Cauchy's  theory  of  functions  of  a  complex  variable.  The  relations  can 
also  be  given  many  very  interesting  physical  interpretations. 

The  simplest  from  a  physical  standpoint  is,  perhaps,  that  in  which  u 
and  v  are  regarded  as  the  component  velocities  in  the  plane  of  x,  y  of  a 
particle  of  a  fluid  in  two-dimensional  motion,  the  particle  in  question  being 
the  particular  one  which  happens  to  be  at  the  point  (x,  y)  at  time  t.  If  u 
and  v  are  independent  of  t  the  motion  is  said  to  be  "steady"  and  a  curve 
along  which  it  is  constant  may  be  regarded  as  a  "stream-line"  or  "line  of 
flow"  of  the  particles  of  fluid.  The  condition  that  a  particle  of  the  fluid 
should  move  along  such  a  line  is,  in  fact,  expressed  by  the  differential 
equations  7  , 

d^=d-y=dt  ......  (B) 

u       v  v    ' 

which  give  vdx  —  udy  =  0, 

that  is  dift  =  0  or  if/  =  constant. 

*  The  reader  is  supposed  to  possess  some  knowledge  of  the  properties  of  analytic  functions. 


Conjugate  Functions  79 

Another  way  of  looking  at  the  matter  is  to  calculate  the  "flux"  across 
any  line  AP  from  right  to  left.  This  is  expressed  by  the  integral 


where  ds  denotes  an  element  of  length  of  AP  and  the  suffix  is  used  to 
indicate  the  point  at  which  \fj  is  calculated.  It  is  clear  from  this  equation 
that  there  is  no  flow  across  a  line  AP  along  which  if/  is  constant. 

The  conjugate  function  <£  is  called  the  "  velocity  potential"  and  was 
first  introduced  by  Euler.  The  curves  on  which  (f>  is  constant  are  called 
"  equipotential  curves."  The  function  </f  is  called  the  stream-function  or 
current  function,  it  was  used  in  a  general  manner  by  Earnshaw. 

It  must  be  understood  that  the  fluid  motion  which  is  represented  by 
such  simple  formulae  is  of  an  ideal  character  and  is  only  a  very  rough 
approximation  to  a  real  motion  of  a  fluid.  A  study  of  this  type  of  fluid 
motion  serves,  however,  as  a  good  introduction  to  the  difficult  mathe- 
matical analysis  connected  with  the  studies  of  actual  fluid  motions.  It  will 
be  worth  while,  then,  to  make  a  few  remarks  on  the  peculiarities  of  this  ideal 
type  of  fluid  motion*. 

In  the  first  place,  it  should  be  noticed  that  the  expression  udx  4-  vdy 
is  an  exact  differential  dcf>,  and  so  the  integral 


I  (udx  +  vdy) 


represents  the  difference  between  the  values  of  <f>  at  the  ends  of  the  path  of 
integration.  If  the  function  (f>  is  one -valued  the  integral  round  a  closed 
curve  is  zero,  but  if  <f>  is  many-valued  the  integral  may  not  vanish.  The 
value  of  the  integral  in  such  a  case  is  called  the  circulation  round  the 
closed  curve.  It  is  different  from  zero  in  the  case  when 

<£  -f  ty  =  i  log  z  =  i  (log  r  +  id) 
and  the  curve  is  a  circle  whose  centre  is  at  the  origin.   In  this  case 

<£  =  _  0,     0  =  log  r, 

and  it  is  easily  seen  that  the  circulation  F  defined  by  the  integral 

f  r  r2zr 

r  =    udx  +  vdy  =  ld<f>  =  -        dO 

J  J  Jo 

is  equal  to  —  2n.  The  fluid  motion  for  which 

<£  -f  ty  =  —  A  log  z, 

where  A  is  a  constant,  is  said  to  be  that  due  to  a  vortex  of  strength  F  when 

ir 

A  is  an  imaginary  quantity  0-  .  If,  on  the  other  hand,  A  is  real,  the  motion 

J-tTT 

is  said  to  be  due  to  a  source  if  —  A  is  positive  and  due  to  a  sink  if  —  A  is 
negative.  The  flow  in  the  last  two  cases  is  radial. 


80  The,"  Classical  Equations 

Since  the  stream  -function  in  the  last  two  cases  is  —  Ad  and  is  not  one- 
valued,  the  flux  across  a  circle  whose  centre  is  at  0  is  —  2-n-A. 

The  flow  due  to  a  vortex,  source  or  sink  at  a  point  other  than  the  origin 
may  be  represented  in  the  same  way  by  simply  interpreting  r  and  6  as 
polar  co-ordinates  relative  to  the  point  in  question. 

Since  the  equations  expressing  u  and  v  in  terms  of  </>  and  $  are  linear, 
the  component  velocities  for  the  flow  due  to  any  number  of  vortices, 
sources  and  sinks  may  be  derived  from  the  complex  potential 

<f>  +  it-  27T  s  («•  -  *&)  log  IX-  x*  +  i(y-  y*)]  > 

where  the  constants  as  ,  j8s  specify  the  strengths  of  the  source  and  vortex 
associated  with  the  point  (xs,  ys).  The  word  source  is  used  here  in  a  general 
sense  to  include  both  source  and  sink. 

One  further  remark  may  be  made  regarding  the  motion  if  we  are 
interested  in  the  career  of  a  particular  particle  of  fluid.  If  x0,  y0  are  the 
initial  co-ordinates  of  this  particle  at  time  t  these  quantities  at  time  t  will 
be  functions  of  x,  y  and  t 

%»  -  /  (x,  y,  0»    2/0  =  g  (x>  y,  0> 

but  functions  of  such  a  nature  that  the  equations  (B)  are  satisfied  when 
xQ  and  y0  are  regarded  as  constant.  We  have  then 

«!/+«!/+  jjf-o,  «*+,  <*+*=,  o    ......  (c.D) 

dx         oy      dt  ox         oy     ot  ' 

and  any  quantity  h  which  can  be  expressed  in  the  form  h  =  F  (XQ  ,  y0)  will 
be  a  solution  of  the  equation 

dh         dh         dh 

a.  +  M  3     +  V  5  -  =  0, 

dt  dx         dy 

and  will  be  constant  throughout  the  motion.  We  shall  write  this  equation 
in  the  form  dhjdt  =  0  and  shall  call  dh/dt  the  complete  time  derivative  of 
h.  When  the  motion  is  steady  we  evidently  have  difj/dt  =  0. 

The  equations  (C)  and  (D)  may  be  solved  for  u  and  v  if  —--       =  1  and 

o  (Xj  yj 


give  expressions  *~ 

which  satisfy  the  equation 


= 

ox     dy 


on  account  of  «  -        >      =  1. 

a  (x,  y)       a  (x,  y) 

This  last  equation  expresses  that  the  area  occupied  by  a  group  of 
particles  remains  constant  during  the  motion.  To  obtain  a  solution  of  this 
equation  we  take  x  and  XQ  as  new  independent  variables,  then 

dy  ^  d  (x,  y)  =  8  (x,  y)  3  (x^y^)  ^  d_(x^  ,J/Q) 
dxQ     d(x,  x0)     3  (x,  *0)    9  (x,  y)       9  (x,  x0)  ' 


Motion  of  a  Fluid  81 

,  dy  dyQ 

and  so  ^-  =  —  /°. 

OXQ  dx 

This  means  that  ydx  —  yQdxQ  is  an  exact  differential  and  so  we  may  write 


where  F  =  F  (x,  x0,  t)  and  t  is  regarded  as  constant.  If,  however,  we  allow 
t  to  vary  and  use  brackets  to  denote  derivatives  when  x}  y  and  t  are 
regarded  as  independent  variables,  we  have 

dyQ  d*F        S2F 

A  —    "U  —   />/ i 

V     1±        ^     'S          'S  I"     <•*»          "  <•>  .    J 

«£  ratoft        OXnCt 


O-i*     i-  ,    -^fog^  \Jfo-;  > 


1  = 


3y/    "  3a:8x0 

_  _        fov 

cte^y     9#9£  "^  9#0 
/3a?0\ 


=  —  u. 

t\cy  / 

O  17f 

Hence  we  may  write  0  =  —    —  and  obtain  a  convenient  expression  for  the 

stream-function. 

Another  physical  interpretation  of  the  functions  <f>  and  $  is  obtained  by 
regarding  <f>  as  the  electric  potential  and  u,  v  as  the  components  of  the 
electric  field  strength  due  to  a  set  of  fictitious  point  charges,  or,  if  we  prefer 
a  three-dimensional  interpretation,  to  a  system  of  uniform  line  charges  on 
lines  perpendicular  to  the  plane  of  x,  y.  The  curves  cf>  =  constant  are  then 
sections  by  this  plane  of  the  equipotential  surfaces  cf>  =  constant,  while  the 
curves  0  =  constant  are  the  "lines  of  force"  in  the  plane  of  x,  y.  For 
brevity  we  shall  sometimes  think  in  terms  of  the  fictitious  point  charges 
and  call  a  curve  <f>  =  constant  an  "equipotential." 

Again,  <f>  may  be  interpreted  as  a  magnetic  potential  of  a  system  of 
magnetic  line  charges  (fictitious  magnetic  point  charges)  or  of  electric 
currents  of  uniform  intensity  flowing  along  wires  of  infinite  length  at  right 
angles  to  the  plane  of  x,  y.  The  curves  $  =  constant  are  again  lines  of  force, 
a  line  of  force  being  defined  by  the  equations 

dx  _dy 

u       v 


82  The  Classical  Equations 

In  all  cases  the  lines  of  force  are  the  orthogonal  trajectories  of  the  equi- 
potentials,  as  may  be  seen  immediately  from  the  relation 


dx  dx      dy  dy  ~~     ' 

which  is  a  consequence  of  Cauchy's  relations. 

For  any  number  of  electric  or  magnetic  line  charges  perpendicular  to 
the  plane  of  x,  y  we  have  by  definition 

<f>  +  iif*  =  22^s  log  [x  -  xs  +  i  (y  -  y,)], 

where  /x8  is  the  density  per  unit  length  of  the  electricity,  or  magnetism  as 
the  case  may  be,  on  the  line  which  passes  through  the  point  (x,  y)\  It  must 
be  understood,  of  course,  that  when  </>  is  the  electric  potential  we  consider 
only  electric  charges  and  when  </>  is  the  magnetic  potential  we  consider  only 
magnetic  charges.  When  the  number  of  terms  in  the  series  is  finite  we  can 

certainly  write  .       .  .       ,  .          , 

J  </»  +  *0=/(x  +  ty)  =/(«), 

where  /  (z)  is  a  function  which  is  analytic  except  at  the  points  z  =  zs  . 

When  in  the  foregoing  equation  p,8  is  regarded  as  a  purely  imaginary 
quantity,  <f>  may  be  interpreted  as  the  magnetic  potential  of  a  system  of 
electric  currents  flowing  along  wires  perpendicular  to  the  plane  of  x,  y. 
If  fjis  =  iC8  the  current  along  the  wire  xs  ,  ys  is  of  strength  C8  and  flows  in 
the  positive  direction,  i.e.  the  direction  associated  with  the  axes  Ox,  Oy  by 
the  right-handed  screw  rule. 

When  a  potential  function  <f>  is  known  it  is  sometimes  of  interest  to 
determine  the  curves  along  which  the  associated  force  (or  velocity)  has 
either  a  constant  magnitude  or  direction.  This  may  be  done  as  follows.  We 

have 


log  (u  -  iv)  =  log/'  (x  +  iy)  =  <X>  +  f¥,  say, 
where  O  =  £  log  (u2  +  v2),     T  =  TT  —  tan*1  (v/u). 

The  curves  O  ==  constant  are  clearly  curves  along  which  the  magnitude 
(u2  +  V2)*  of  the  force  or  velocity  is  constant,  while  Y  =  constant  is  the 
equation  of  a  curve  along  which  the  direction  of  the  force  is  constant.  The 
functions  <t>  and  T  are  clearly  solutions  oi  Laplace's  equations,  i.e. 

320     32O  _ 

dx2  +  dy2  ~ 

A  function  O  which  satisfies  this  equation  is  called  a  logarithmic 
potential  to  distinguish  it  from  the  ordinary  Newtonian  potential  which 
occurs  in  the  theory  of  attractions.  The  electric  and  magnetic  potentials  of 
line  charges  are  thus  logarithmic  potentials. 


Two-Dimensional  Stresses  83 

A  logarithmic  potential  <D  is  said  to  be  regular  in  a  domain  D  if 


ao    ao 

'  >  '          2' 


are  continuous  functions  of  x  and  y  for  all  points  of  D.    If  D  is  a  region 
which  extends  to  infinity  it  is  further  stipulated  that 

lim  <f>  (x,  y)  =  (7,     lim  r  -  ^  =  lim  r  ~  -  =  0, 

r  —  >  oo  r  —  >•  oo       0«£        /  —  >  oo      0y 

(r2  -  a:2  +  y*), 

where  C  is  a  finite  quantity  which  may  be  zero.  In  this  sense  the  potential 
of  a  single  line  charge  is  not  regular  at  infinity. 

Still  another  physical  interpretation  of  conjugate  functions  is  obtained 
by  writing  Y  y       JL      y        y       ,/, 

Ax     =       —        I    y     =     0,  Ay     =        I    X     =      </f. 

Cauchy's  relations  then  give 


These  are  the  equations  for  the  equilibrium  of  an  elastic  solid  when  there 
are  no  body  forces  and  the  stress  is  two-dimensional.  The  quantities 
(Xx,  Xv)  are  interpreted  as  the  component  stresses  across  a  plane  through 
(x,  y)  perpendicular  to  the  axis  of  x,  while  (Yx,  Yv)  are  the  component 
stresses  across  a  plane  perpendicular  to  the  axis  of  y.  The  relation  Xv  =  Yx 
is  quite  usual  but  the  relation  Xx  +  Yy  —  0  indicates  that  the  distribution 
of  stress  is  of  a  special  character.  A  stress  system  satisfying  this  condition 
can,  however,  be  obtained  by  writing 

*„--  rv=£(«2-»2),     Yx  =  Xv=*-uv, 
for  these  equations  give 

SXX     dXv         /du     dv\         /dv     du 


The  fact  that  the  various  potentials  <f>  and  0  which  have  been  considered 
so  far  are  solutions  of  Laplace's  equation 


is  a  consequence  of  the  circumstance  that  they  have  been  defined  as  sums 
of  quantities  that  are  individually  solutions  of  this  equation.  No  physical 
principle  has  been  used  except  a  principle  of  superposition  which  states 
that  when  the  individual  terms  give  quantities  with  a  physical  meaning, 

6-2 


84  The  Classical  Equations 

the  sum  will  give  a  quantity  with  a  similar  physical  meaning.  In  the 
analysis  of  many  physical  problems  such  a  superposition  of  individual 
effects  is  not  strictly  applicable,  for  the  sources  of  a  disturbance  cannot  be 
supposed  to  act  independently,  each  source  may,  in  fact,  be  modified  by 
the  presence  of  the  others  or  may  modify  the  mode  of  propagation  of  the 
disturbance  produced  by  another.  Such  interactions  will  be  left  out  of 
consideration  at  present,  for  our  aim  is  not  to  formulate  at  the  outset  a 
complete  theory  of  physical  phenomena  but  to  gradually  make  the  student 
familiar  with  the  mathematical  processes  which  have  been  used  successfully 
in  the  gradual  discovery  of  the  laws  of  physical  phenomena. 

In  applied  mathematics  the  student  has  always  found  the  formulation 
of  the  fundamental  equations  of  a  problem  to  be  a  matter  of  some  difficulty. 
Some  men  have  been  very  successful  in  formulating  simple  equations  be- 
cause, by  a  kind  of  physical  instinct,  they  have  known  what  to  neglect.  The 
history  of  mathematical  physics  shows  that  in  many  cases  this  so-called 
physical  instinct  is  not  a  safe  guide,  for  terms  which  have  been  neglected 
may  sometimes  determine  the  mathematical  behaviour  of  the  true  solution. 
In  recent  years  the  tendency  has  been  to  try  to  work  with  partial  differential 
equations  and  their  solutions  without  the  feeling  of  orthodoxy  which  is 
created  by  a  derivation  of  the  equations  that  is  regarded  for  the  time  being 
as  fully  satisfactory.  The  mathematician  now  feels  that  it  is  only  by  a 
comparison  of  the  inferences  from  his  equations  with  the  results  of  ex- 
periment and  the  inferences  from  slightly  modified  equations  that  he  can 
ascertain  whether  his  equations  are  satisfactory  or  not.  In  the  present 
state  of  physics  the  formulation  of  equations  has  not  the  air  of  finality 
that  it  had  a  few'  years  ago. 

This  does  not  mean,  however,  that  the  art  of  formulating  equations 
should  be  neglected,  it  means  rather  that  mathematicians  should  also 
include  amongst  their  special  topics  of  study  the  processes  which  lead  to 
the  most  interesting  partial  differential  equations  of  physics.  These  pro- 
cesses are  of  various  kinds.  Besides  the  process  of  elimination  from  equa- 
tions of  the  first  order  there  are  the  methods  of  the  Calculus  of  Variations 
and  methods  which  depend  upon  the  use  of  line,  surface  and  volume 
integrals.  Mathematically,  the  direct  process  of  elimination  is  the  simplest 
and  will  be  given  further  consideration  in  §  1-82. 

/ 

§  1-72.  Geometrical  properties  of  equipotentials  and  lines  of  force.  When 
the  potential  </>  is  a  single-valued  function  of  x  and  y  there  cannot  be  more 
than  one  equipotential  curve  through  a  given  point  P  in  the  (x,  y)  plane. 
An  equipotential  curve  <£  =  </>0  may,  however,  cross  itself  at  a  point  and 
have  a  multiple  point  of  any  order  at  a  point  PQ.  In  such  a  case  the 
tangents  at  the  multiple  point  are  arranged  like  the  radii  from  the  centre 
to  the  corners  of  a  regular  polygon.  To  see  this,  let  us  take  the  origin  at  P0, 


Method  of  Inversion  85 

then  the  terms  of  lowest  degree  in  the  Taylor  expansion  of  </>  —  <£0  are  of 

yP  cnenla-  (x  4-  iy)n  4-  cne~nt(l  (x  —  iy)n, 

where  n  is  an  integer  and  c  and  a  are  constants.  In  polar  co-ordinates 
x  =  r  cos  9,  y  =  r  sin  #,  these  terms  become 

2cnrn  cos  ^  (0  4-  «), 

and  the  directions  of  the  n  tangents  are  given  by  cos  n  (6  -f  «).  The  possible 
values  of  n  (9  4-  a)  are  thus  ?r/2,  37r/2,  ...  (n  —  J)  TT,  the  angle  between  con- 
secutive tangents  being  TT/U. 

Since  cos  n  (9  4-  a)  is  positive  for  some  values  of  6  and  negative  for 
others,  the  function  <f>  cannot  have  a  maximum  or  minimum  value  at  a 
point,  for  this  point  may  be  chosen  for  origin  and  the  expansion  shows  that 
there  are  points  in  the  immediate  neighbourhood  of  the  origin  for  which 
</>  >  (f)Q  ,  and  also  points  for  which  <£  <  </>0  . 

By  means  of  the  transformation 
x'  -  iy'  =  k*  (x  4-  iy)-1, 
x'  -f  ^y  =  &2  (x  -  iy)~l, 
which   represents    an    inversion    with 
respect  to  a  circle  of  radius  k  and  centre 
at  the  origin,  an  equipotential  curve  of 
a  system  of  line  charges  is  transformed  1S' 

into  an  equipotential  curve  of  another  system  of  line  charges. 

In  polar  co-ordinates  we  have 

r'  =  *Yr>     9'  =  °- 
If  in  Fig.  10  Q  corresponds  to  P  and  B  to  A,  we  have 

Rjr  =  B'/b, 
where  AP  =  R,  OP  =  r,  BQ  -  R',  OB  =  b. 

For  a  number  of  points  A  and  the  corresponding  points  B 

S/t.  log  (R,/r)  =  SMs  (log  «'  -  log  b). 
An  equipotential  system  of  curves  represented  by  the  equation 


is  thus  transformed  into  an  equipotential  system  represented  by  the 
equation 


=  C  _  ^^  log  6. 

A  line  charge  at  B  is  seen  to  correspond  to  a  line  charge  of  equal  strength 
at  A  and  another  one  of  opposite  sign  at  0  which  may  be  supposed  to 
correspond  to  a  line  charge  at  infinity  sufficient  to  compensate  the  charge 
at  B. 

Ap  equipotential  curve  with  a  multiple  point  at  O  inverts  into  an  equi- 
potential which  goes  to  infinity  in  the  directions  of  the  tangents  at  the 


86  The  Classical  Equations 

multiple  point.  This  indicates  that  the  directions  in  which  an  equipotential 
goes  to  infinity  are  parallel  to  the  radii  from  the  centre  to  the  corners  of 
a  regular  polygon. 

In  the  simple  case  of  two  equal  line  charges  at  the  points  (c,  0),  (—  c,  0) 
the  equipotentials  are 

log  Rl  +  log  R2  =  constant, 
or  R^2  =  a2, 

where  a  is  constant  for  each  equipotential.  These  curves  are  Cassinian 
ovals  with  the  polar  equation 

r4  -f  c4  -  2r2c2  cos  20  =  a4. 

When  a  =  c  we  obtain  the  lemniscate  r2  =  c2  cos  26  with  a  double 
point  at  the  origin.  The  tangents  at  the  double  point  are  perpendicular. 

Inverting  we  get  the  equipotentials  for  two  equal  line  charges  of 
strength  +  1  at  the  points  (6,  0),  (-  6,  0),  where  be  =  k2  and  a  line  charge 
of  strength  —  2  at  the  origin.  The  equipotentials  are  now 

log  RI  +  log  jR2'  -  2  log  r'  =  constant, 
or  c*R/jR2'  =  aV2. 

Dropping  the  primes  we  have  the  polar  equation 
C2  (r4  +  C4  _  2r2c2  cos  20)  -  a*r4 

of  a  system  of  bicircular  quartic  curves.  When  a  =  c  we  obtain  the  rect- 
angular hyperbola  r2  cos  20  =  c2  which  is  the  inverse  of  the  lemniscate. 
The  rectangular  hyperbola  goes  to  infinity  in  two  perpendicular  directions. 

It  is  easily  seen  that  lines  of  force  invert  into  lines  of  force.  In  Fig.  10, 
if  we  denote  the  angles  POA,  PAB,  QBO  by  d,  Q  and  0'  respectively,  we 
have  the  relation  _ 

Hence  the  lines  of  force  represented  by  the  equation 

S  fj,s  ©/  =  constant 
transform  into  the  lines  of  force  represented  by  the  equation 

2/A,05  —  02/is  =  constant. 

In  particular,  the  lines  of  force  of  two  equal  line  charges 
©i  H-  ©2  =  constant, 

being  rectangular  hyperbolas,  invert  into  the  family  of  lemniscates  repre- 
sented  by  _ 


and  these  are  the  lines  of  force  of  two  equal  line  charges  of  strength  -f  1 
and  a  single  line  charge  of  strength  -  2  at  0. 

At  a  point  of  equilibrium  in  a  gravitational,  electrostatic  or  magnetic 
field,  the  first  derivatives  of  the  potential  vanish  and  so  the  equipotential 
curve  through  the  point  has  a  double  point  or  multiple  point.  A  similar 


Lines  of  Force  87 

remark  applies  to  a  curve  ifj  =  constant,  but  this  curve  cannot  strictly  be 
regarded  as  a  single  line  of  force  for,  if  we  consider  any  branch  which  passes 
through  the  point  of  equilibrium  without  change  of  direction,  the  force  is 
in  different  directions  on  the  two  sides  of  the  point  of  equilibrium  and  the 
neighbouring  linqs  of  force  avoid  the  point  of  equilibrium  by  turning  through 
large  angles  in  a  short  distance.  This  is  exemplified  in  the  case  of  two  equal 
masses  or  charges  when  the  equipotentials  are  Cassinian  ovals  which  include 
a  lemniscate  with  a  double  point  at  the  point  of  equilibrium.  The  lines  of 
force  are  then  rectangular  hyperbolas,  the  system  including  one  pair  of 
perpendicular  lines  which  cross  at  the  point  of  equilibrium. 

In  plotting  equipotential  curves  and  lines  of  force  for  a  given  system  of 
line  charges  it  is  very  useful  to  know  the  position  of  the  points  of  equi- 
librium, since  the  properties  just  mentioned  can  be  employed  to  indicate 
the  behaviour  of  the  lines  of  force.  At  a  point  of  stagnation  in  an  irrota- 
tional  two-dimensional  flow  of  an  inviscid  fluid  the  component  velocities 
vanish  and  so  the  first  derivatives  of  the  velocity  potential  and  stream- 
function  are  zero.  The  properties  of  the  equipotentials  and  stream-lines  at 
a  point  of  stagnation  are,  then,  similar  to  those  of  equipotential  and  lines 
of  force  at  a  point  of  equilibrium.  There  is,  however,  one  important 
difference  Between  the  two  cases.  In  the  electric  problem  the  field  is  often 
bounded  by  a  conductor,  i.e.  an  equipotential  surface,  while  in  the  hydro- 
dynamical  problem  the  field  of  flow  is  generally  bounded  by  some  solid 
body  whose  profile  in  the  plane  z  =  0  is  a  stream-line.  A  point  of  stagnation 
frequently  lies  on  the  boundary  of  the  body  and  two  coincident  stream- 
lines may  be  supposed  to  meet  and  divide  there,  running  round  the  body 
in  opposite  directions  and  reuniting  at  the  back  of  the  body  when  the 
profile  is  a  simple  closed  curve. 

A  point  on  a  conductor  may  be  a  point  of  equilibrium  if  the  conductor's 
profile  is  a  curve  with  a  double  point  with  perpendicular  tangents  or  if  it 
consists  of  two  curves  cutting  one  another  orthogonally  at  all  their 
common  points.  It  should  be  remarked,  however,  that  the  force  at  a  double 
point  may  be  either  zero  or  infinite ;  it  is  zero  when  the  double  point  repre- 
sents a  pit  or  dent  in  the  curve,  but  is  infinite  when  the  double  point 
represents  a  peak.  This  may  be  exemplified  by  the  equations  <f>  =  x2  —  y2, 
iff  =  2xy.  If  the  field  lies  in  the  region  x  >  0,  y2  <  x2,  the  force  is  zero  at 
0  and  there  is  a  single  line  of  force  through  0,  namely,  y  =  0  (Fig.  11). 
If,  on  the  other  hand,  the  field  is  outside  the  region  x  <  0,  x2  >  y2,  and 
,  _  x2-y2  _  2xy 


the  force  at  the  origin  is  infinite  for  most  methods  of  approach  and  there 
are  three  lines  of  force  through  the  origin  (Fig.  12). 

Similarly,  when  two  conductors  meet  at  any  angle  less  than  ?r,  but  a 
submultiple  of  TT,  the  angle  being  measured  outside  the  conductor.  The 


88  The  Classical  Equations 

point  of  intersection  is  a  point  of  equilibrium  and  we  have  the  approximate 

<£  =  2cnrn  cos  n  (6  -f  «) 


expression 


of  Force 


Line 
of  Force 

Fig.  12. 

for  the  value  of  </>  in  the  neighbourhood  of  the  point,  the  equation  of  the 
conductor  in  the  neighbourhood  of  the  point  being  n  (9  -f  «)  =  ±  n/2  and 

the  field  being  in  the  region  —  9  <  n  (9  -f  a)  <  ~ .  The  angle  is  in  this  case 

£  £ 

7T/n  and  the  radial  force  ^  varies  initially  according  to  the  (n  —  l)th  power 

of  the  distance  as  a  point  recedes  from  the  position  of  equilibrium. 
The  corresponding  approximate  expression  for  «/r  is 

ifj  =  2cnrn  sin  n  (6  +  a) 

and  there  is  a  single  line  of  force  9  =  —  a  which  lies  within  the  field,  this 
being  its  equation  in  the  immediate  neighbourhood  of  the  point  O. 

There  is  another  simple  transformation  which  is  sometimes  useful  for 
deriving  the  equipotentials  and  lines  of  force  of  one  set  of  line  charges  from 
those  of  another.  This  is  the  transformation 

z'  =  z  +  a2/z 
which  gives  two  values  of  z  for  each  value  of  z' .  Let  these  be  z  and  z,  then 


zz  —  a* 


Similarly,  let  z1  and  zl  correspond  to  z/,  then 


z'  -  z/  -  z  -  zl  -f  a*/z  -  a2/*!  =  (z  -  zj  (1  -  «2 

=   (Z  -  2J  (1   -  V)  =   (Z  -  Z,)  (Z  -  Zj/Z. 

Taking  the  moduli  we  obtain  a  relation 


where 


r    =     z  -  z 


r  = 


Geometrical  Properties  89 

Similarly,  if  z2  and  z2  correspond  to  z2'  we  have,  with  a  similar  notation, 

<  =  r2r2fr, 

j  r2x     r2r2 

and  so  -~  =  -~~  . 

ri      Wi 

The  transformation  thus  enables  us  to  derive  the  equipotentials  for  four 
charges  (1,  1,  —  1,  —  1)  from  the  equipotentials  for  two  charges  (1,  —  1) 
and  a  similar  remark  holds  for  the  lines  of  force,  as  may  be  seen  by  equating 
the  arguments  on  the  two  sides  of  the  equation 


This  is  just  one  illustration  of  the  advantages  of  a  transformation. 

A  general  theory  of  such  transformations  will  be  developed  in 
Chapter  III. 

Some  geometrical  properties  of  equipotential  curves  and  lines  of  force 
may  be  obtained  by  using  the  idea  of  imaginary  points.  The  pair  of  points 
with  co-ordinates  (a  ±  ij8,  b  T  ia)  are  said  to  be  the  anti-points  of  the  pair 
with  co-ordinates  (a  ±  a,  b  ±  /?),  the  upper  or  lower  sign  being  taken 
throughout.  Denoting  the  two  pairs  by  Fl  ,  F2  ;  Sl  ,  S2  respectively,  we  can 
say  that  if  S1  and  S2  are  the  real  foci  of  an  ellipse,  then  Ft  and  F2  are  the 
imaginary  foci.  Fl  and  F2  can  also  be  regarded  as  the  imaginary  points  of 
intersection  of  the  coaxial  system  of  circles  having  8l  and  S2  as  limiting 
points. 

If  the  co-ordinates  of  F±  and  F2  are  (xl9  yj,  (x2,  y2)  respectively  arid 
those  of  Slt  S2  are  (£19  T?^,  (£2,  7?2)  respectively,  we  have 

xi  +  iyi  =  a  +  *  4-  a  -f  ij8  =  &  +  iifr, 
x\  -  iy\  =  a  ~  ib  -  a  +  ip  =  ^2  -  i^2, 
xz  +  iy*  =  a  +  ib  -  «  -  *j3  =  ^2  -f  i^2, 
^2  ~  *2/2  =  a  —  i&  +  a  —  *'j8  =  f  !  —  i-^!  . 
If  now  i£  -f-  iv  —  f  (x  +  iy),  u  —  iv  —  f  (x  —  iy), 

and  $!  ,  $2  lie  on  a  curve  u  =  constant,  we  have 


The  foregoing  relations  now  show  that 

/  te  +  iyi)  +  f  fa  -  iy*)  =  /  (*2  + 

and  this  means  that  Fl9  F2  lie  on  a  curve  v  =  constant. 

When  the  imaginary  points  on  a  curve  v  =  constant  admit  of  a  simple 
geometrical  representation  or  description,  the  foregoing  result  may  be 
sometimes  used  to  find  the  curves  u  —  constant.  If  the  curve  v  =  constant 
is  a  hyperbola,  the  imaginary  points  in  which  a  family  of  parallel  lines  meet 
the  curve  have  geometrical  properties  which  are  sufficiently  well  known 
to  enable  us  to  find  the  anti-points  of  each  pair  of  points  of  intersection. 


90  The  Classical  Equations 

These  anti-points  lie  on  a  confocal  ellipse  which  is  a  curve  of  the  family 
u  =  constant.  By  taking  lines  in  different  directions  the  different  ellipses 
of  the  family  u  —  constant  are  obtained.  Similarly,  by  taking  a  set  of 
parallel  chords  of  an  ellipse  and  the  anti-points  of  the  two  points  of  inter- 
section of  each  chord,  it  turns  out  that  these  anti-points  all  lie  on  a  confocal 
hyperbola,  and  by  taking  families  of  lines  with  different  directions  the 
different  hyperbolas  of  the  confocal  family  may  be  obtained. 

In  this  case  the  relations  are  particularly  simple.  In  the  general  case 
when  one  curve  of  the  family  u  =  constant  is  given  there  will  be,  pre- 
sumably, a  family  of  lines  whose  imaginary  intersections  with  this  curve 
are  pairs  of  points  with  anti-points  lying  on  one  curve  of  the  family  v  =  con- 
stant, but  these  lines  cannot  be  expected,  in  general,  to  be  parallel,  and  a 
simple  description  of  the  family  is  wanting. 

EXAMPLES 

1.  If  a  family  of  circles  gives  a  set  of  equipotential  curves,  the  circles  are  either  con- 
centric or  coaxial. 

2.  Equipotentials  which  form  a  family  of  parallel  curves  must  be  either  straight  lines 
or  circles. 

[Proofs  of  these  propositions  will  be  found  in  a  paper  by  P.  Franklin,  Journ.  of  Math. 
and  Phys.  Mass.  Inst.  of  Tech.  vol.  vi,  p.  191  (1927).] 

§  1-81.  The  classical  partial  differential  equations  for  Euclidean  space. 
Passing  now  to  the  consideration  of  some  partial  differential  equations  in 
which  the  number  of  independent  variables  is  greater  than  two  we  note 
here  that  the  most  important  equations  are  Laplace's  equation 

32F     32F     32F      „ 


the  wave-equation  ^v     92F     92F      1  92F 

"S*»  +  ay«  +  a*'2  ^c2  w 

the  equation  of  the  conduction  of  heat 


the  equation  for  the  conduction  of  electricity 

dE 

^     .......  (D) 


and  the  wave-equation  of  Schrodinger's  theory  of  wave  -mechanics.  This 
last  equation  takes  many  different  forms  and  we  shall  mention  here  only 
the  simple  form  of  the  equation  in  which  the  dependence  of  0  on  the  time 
has  already  been  taken  into  consideration.  The  reduced  equation  is  then 


. 
8?+^(        ^=  '  ......  (  } 

where  F  is  a  function  of  x,  y  and  z  and  E  is  a  constant  to  be  determined. 


Laplace's  Equation  91 

In  these  equations  K  represents  the  diffusivity  or  thermometric  con- 
ductivity of  the  medium,  K  the  specific  inductive  capacity,  /x  the  per- 
'meability,  and  a  the  electric  conductivity  of  the  medium.  The  quantities 
c  and  h  are  universal  constants,  c  being  the  velocity  of  light  in  vacuum 
and  h  being  Planck's  constant  which  occurs  in  his  theory  of  radiation. 

Laplace's  equation,  which  for  brevity  may  be  written  in  the  form 

V2V  =  0, 

may  be  obtained  in  various  ways  from  a  set  of  linear  equations  of  the  first 
order.  One  set, 

dv          dv         dv    sx    dY    dz 
x=s*'    Y=dy>  z  =  ~dz'    a*+ay  +  al  =  0' (F) 

occurs  naturally  in  the  theory  of  attractions,  V  being  the  gravitational 
potential  and  X,  Y,  Z  the  components  of  force  per  unit  mass.  The  last 
equation  is  then  a  consequence  of  Gauss's  theorem  that  the  surface  integral 
of  the  normal  force  is  zero  for  any  closed  surface  not  containing  any 
attracting  matter. 

The  same  equations  occur  also  in  hydrodynamics,  the  potential  V  being 
replaced  by  the  velocity  potential  <£  and  the  quantities  X,  Y,  Z  by  the 
component  velocities  u,  v,  w.  The  equation  is  then  the  equation  of  con- 
tinuity of  an  incompressible  fluid. 

The  electric  and  magnetic  interpretations  of  X ,  Y,  Z  and  V  are  similar 
to  the  gravitational  except  that  the  electric  (or  magnetic)  potential  is 
usually  taken  to  be  —  V  when  X ,  Y ,  Z  are  the  force  intensities. 

As  in  the  two-dimensional  theory,  Laplace's  equation  is  satisfied  by  the 
potential  V  because  by  the  principle  of  superposition  V  is  expressed  as  the 
sum  of  a  number  of  elementary  potentials  each  of  which  happens  to  be  a 
solution  of  Laplace's  equation,  the  elementary  potential  being  of  type 

V=[(x-  x')*  +(y-  y')2  +  (a  -  z')T}  =  l/S. 

When  V  is  interpreted  as  the  electrostatic  potential  this  elementary 
potential  is  regarded  as  that  of  a  unit  point  charge  at  the  point  (#',  y* ',  z') ; 
when  V  is  interpreted  as  a  magnetic  potential  the  elementary  potential  is 
that  of  a  unit  magnetic  pole.  In  the  theory  of  gravitation  the  elementary 
potential  is  that  of  unit  mass  concentrated  at  the  point  (#,  yy  z).  A  more 
general  expression  for  a  potential  is 

V  =  2m5  [(x  -  *.)«  +  (y  -  ys)*  +  (z  -  ».)•]-*, 

where  the  coefficient  ms  is  a  measure  of  the  strength  of  the  charge,  pole  or 
mass  concentrated  at  the  point  (x8,  y8,  zs).  If  we  write  <f>  in  place  of  V, 
where  <f>  is  a  velocity  potential  for  a  fluid  motion  in  three  dimensions,  the 
elementary  potential  is  that  of  a  source  and  the  coefficient  ms  can  be 
interpreted  as  the  strength  of  the  source  at  (x3,  y8,  zs).  Sources  and  sinks 


92  The  Classical  Equations 

are  useful  in  hydrodynamics  as  they  give  a  convenient  representation  of 
the  disturbance  produced  by  a  body  when  it  is  placed  in  a  steady  stream. 

§  1-82.   Systems  of  partial  differential  equations  of  the  first  order  which 
lead  to  the  classical  equations.   When  we  introduce  algebraic  symbols 

x  ~  dx '        v  ~~  dy '        z  ~~  dz 

for  the  differential  operators  the  equations  (F)  of  §  1-81  become 

X  -  Dx  V  =  0, 
7  -  Dy  V  =  0, 

z  -  A  v  =  o, 


and  the  algebraic  eliminant 


1 

0 

0 

-A 

0 

1 

0 

-A 

0 

0 

1 

-A 

D 

A 

X) 

0 

-  0 


is  simply  Z>x2  -f  Z>v2  4-  A2  =  0. 

If,  on  the  other  hand,  we  consider  the  set  of  equations 

dw      dv      ds  _ 
dy      dz      dx        ' 

du     dw      ds 

—  -  ,  _        r=    {J 

dz  ox  dy 

dv  du  ds  _ 

3#  3?/  82  ~    ' 

du  dv  dw 

which  give  V2u  =  V2v  =  V2w;  =  V2*s  =  0, 

the  corresponding  algebraic  equations 


.(G) 


give  the  eliminant 


-  AM  + 

Af 

Dvw- 

Dxs  =  0, 

A*  =  °> 
A«  =  o, 

AM  + 

A«  + 

Dzw 

=  0 

0 

-A 

Dy 

-A 

=  o, 

A 

0 

-A 

-A 

-A 

-DX 

0 

-A 

A 

/>« 

A 

0 

which  is  equivalent  to 


+ 


+ 


•(H) 


(I) 


Elastic  Equilibrium  93 

These  examples  show  that  the  problem  of  finding  a  set  of  linear  equa- 
tions of  the  first  order  which  will  lead  to  a  given  partial  differential  equation 
of  higher  order  admits  a  variety  of  solutions  which  may  be  classified  by 
noting  the  power  of  the  complete  differential  operator  (in  this  case  (V2)2) 
which  is  represented  by  the  algebraic  eliminant  written  in  the  form  of  a 
determinant. 

It  is  known  that  Laplace's  equation  also  occurs  in  the  theory  of  elas- 
ticity. If  u,  v,  w  denote  the  components  of  the  displacements  and  Xx,  Yy, 
Zz,  Yz,  Zx,  Xy  the  component  stresses  the  equations  for  the  case  of  no 
body  forces  are 


.(J) 


*   1 

a# 

v 

ay  + 

u^\.  z 

Tz'~ 

o, 

dYx 

dYy  , 

dYz 

A 

dx 

dy     ' 

dz 

U) 

™*4 

dZv 

SZZ_ 

0, 

and  if  the  substance  is  isotropic  the  relations  between  stress  and  strain 
take  the  form  ~ 


=  AA 


.(K) 


where 


V  r/  [  ^™    i 

7--z'  =  *(%  +  &)• 


du     dw* 

+  dx 

du 


.  _  du      dv     dw 
dx     dy      dz  ' 


The  equations  obtained  by  eliminating  Xx,  Yy,  Zz,  Yz,  Zx,  Xv  are 

=  (A  +  ^)~, 


=  (A  +  11.) ' 


and,  except  in  the  case  when  A  +  2/*  =  0,  a  case  which  is  excluded  because 
A  and  fj.  are  positive  constants  when  the  substance  is  homogeneous,  these 


94  The  Classical  Equations 

equations  imply  that  A  is  a  solution  of  Laplace's  equation.  The  algebraic 
eliminant  is  in  this  case 

(D,2  +  A,2  +  A2)3  =  0.  ......  (L) 

It  is  easily  seen  that  the  quantities  XX9  Yy,  Zz,  7C,  Zx,  Xy,  u,  v,  w  are 
all  solutions  of  the  equation  of  the  fourth  order 

V2V2w  =  0, 
i.e.  V4^  =  0, 

which  may  be  called  the  elastic  equation.  The  algebraic  equation  obtained 
by  eliminating  the  twelve  quantities  Xx,  Xy,  Xz,  YX)  Yy,  Yz,  Zx,  Zy,  Zz, 
u,  v,  w  from  the  twelve  equations  (J)  and  (K)  by  means  of  a  determinant 
is  also  equivalent  to  (L). 

The  question  naturally  arises  whether  as  many  as  four  equations  are 
necessary  for  'the  derivation  of  Laplace's  equation  from  a  set  of  equations 
of  the  first  order.  The  answer  seems  to  be  yes  or  no  according  as  we  do 
or  do  not  require  all  the  quantities  occurring  in  the  linear^quations  of  the 
first  order  to  be  real.  Thus,  if  we  write  U  =  u  —  iv,  V  =  w  +  is,  where 
u,  v,  w,  s  are  the  quantities  satisfying  the  equations  (H),  it  is  easily  seen 

that  du    .du   dv    A    dv    .dv    du    „ 

.  1        4    .  ___          I  __    -     I  I  _     _     n  _     A 

~liT~  I          V       *\  I  ?N  -      VJ  ~f\  V       f*      ~      -     ~^.      ~      -      U. 

dx         oy      dz  ex         cy       dz 

and  these  equations  imply  that 

=  0,     V2F=  0. 


The  algebraic  eliminant  is  in  this  case  simply  (G). 

It  should  be  noticed  that  if  we  write  ict  in  place  of  y  the  two-dimensional 
wave-equation 


may  be  derived  from  the  two  equations 

dU     dV     13^_0      ?Z_^_l?Z_n 

dx  +  dz  +  c  ~dt  ~    '     dx      dz      c~dt~ 

which  have  real  coefficients.  The  wave-equation  (B)  may  also  be  derived 
from  two  linear  equations  of  the  first  order 

dU       .dU_dV      IdV 
dx  +  l  dy  ~  ~fa  +  c  dty 

w__idv^i  su_du 

dx         dy  "~  c  dt       dz  ' 

but  in  this  case  the  coefficients  are  not  all  real.  The  algebraic  eliminant  is 
in  this  case  simply 

*  c2  (IV  +  Dv2  -f  A2)  -  A2  =  0. 


Equations  leading  to  the  Wave-Equation  95 

To  obtain  the  wave-equation  from  a  set  of  linear  equations  of  the  first 
order  with  only  real  coefficients  we  may  use  the  set  of  eight  linear  equations, 


dy 

dz~ 

'ds 

dx' 

dy 

dz 

~dx 

ds' 

da 

dy_ 

dY 

dT 

dx 

dz 

_dr 

dp 

dz 

dx 

'ds 

dy' 

'dz 

dx 

dy 

ds' 

dp 

da_ 

dZ 

dT 

dY 

dx 

dr 

dy 

dx~ 

dy~ 

ds 

dz' 

'dx 

dy 

~dz 

ds' 

X     i 

57_ 

dT 

dz 

da 

dp 

dr 

dy 

Jx  +  ' 

d~y~ 

ds 

dz' 

dx4 

'dy 

~ds 

~dz' 

in  which  for  convenience  s  has  been  written  in  place  of  ct. 

These  equations  imply  that  X ,  Y,  Z,  T,  a,  j3,  y  and  r  are  all  solutions 
of  the  wave-equation.  The  algebraic  eliminant  is  now 

0*  =  (Dx*  +  Dv*  +  A2  -  A2)4  =  0. 

If  in  the  foregoing  equations  we  put  T  =  r  =  0  we  obtain  a  set  of 
equations  very  similar  to  that  which  occurs  in  Maxwell's  electromagnetic 
theory.  The  eight  equations  may  be  divided  into  two  sets  of  four  and  an 
algebraic  eliminant.  may  be  obtained  by  taking  three  equations  from  each 
set  and  eliminating  the  six  quantities  X,  7,  Z,  «,  /?,  y.  There  are  altogether 
sixteen  possible  eliminants  but  they  are  all  of  type  02L  =  0,  where  the 
last  factor  L  is  obtained  by  multiplying  a  term  from  the  first  of  the  two 

D.    Dv    A    D. 
Dx    Dy    Dz    D8 
by  a  term  from  the  second. 

§1-91.  Primary  solutions.  Let/  (&,  £2,  ...  £w)  be  a  homogeneous  poly- 
nomial of  the  degree  in  its  m  arguments  & ,  £2  >  •  •  •  £w  an<^  ^e^  each  of  the 
quantities  D8  that  is  used  to  denote  an  operator  d/dx8  be  treated  as  an 
algebraic  quantity  when  successive  operations  are  performed.  The  equation 

/(A,A,-A»)*=o  (A) 

is  then  a  linear  homogeneous  partial  differential  equation  of  a  type  which 
frequently  occurs  in  physics.   An  equation  such  as 

Dfw  =  D2w 
may  be  included  among  equations  of  the  foregoing  type  by  writing 

u  =  e** .  w, 
and  noting  that  u  satisfies  the  equation 

(A2  ~  A  A)  u  =  0. 
A  solution  of  the  form 


96  The  Classical  Equations 

in  which  0X,  02,  ...  0,  are  particular  functions  of.a^,  x2,  x3,  ...  xm,  and  F  is 
an  arbitrary  function  of  the  parameters  019  02,  03,  ...  0>s.,  will  be  called  a 
primary  solution.  An  arbitrary  function  will  be  understood  here  to  be  a 
function  which  possesses  an  appropriate  number  of  derivatives  which  are 
all  continuous  in  some  region  R.  Such  a  function  will  be  said  to  be  con- 
tinuous (/),  n)  when  derivatives  up  to  order  n  are  specified  as  continuous. 

It  can  be  shown  that  the  general  equation  (A)  always  possesses  primary 
solutions  of  type  u  =  F  (0),  ......  (B) 

where  6  ==  c,^  4-  c2x2  -f  ...  cmxm,  ......  (C) 

and  ct,  c2,  ...  cm  are  constants  satisfying  the  relation 

/(q,^,  ...CTO)-=  0.  ......  (D) 

This  relation  may  be  satisfied  in  a  variety  of  ways  and  when  a  para- 
metric representation  ~     , 


c   = 


cro  ~   C'm  (ai>  «2>   •••  am-2 

is  known  for  the  cD-ordinates  of  points  on  the  variety  whose  equation  is 
represented  by  (D),  the  formulae  (B)  and  (C)  will  give  a  family  of  primary 
solutions. 

When  ra  =  2  there  is  generally  no  family  of  primary  solutions  but 
simply  a  number  of  types,  thus  in  the  case  of  the  equation 

(A2  -  A2)  u  =  o 

there  are  the  two  types 

u  =  F  (^  -f  a;2),     te  =  J7  (^  -  x2). 

Primary  solutions  may  be  generalised  by  summing  or  integrating  with 
respect  to  a  parameter  after  multiplication  by  an  arbitrary  function  of 
the  parameter.  Thus  in  the  case  of  Laplace's  equation  we  have  a  family  of 
primary  solutions  V  =  F  (0)  .  G  (a),  where 

0  =  z  -f  ix  cos  a  -f  iy  sin  «, 

and  a  is  an  arbitrary  parameter.  Generalisation  by  the  above  method  leads 
to  a  solution  which  may  be  further  generalised  by  summation  over  a 
number  of  arbitrary  functional  forms  for  F  (0)  and  G  (a)  and  we  obtain 
Whittaker's  solution  * 

f2ir 

V  =         W  (z  -f  ix  cos  a  -f  iy  sin  a,  a)  da, 
.'o 

which  may  also  be  obtained  directly  by  making  the  arbitrary  function  F 
a  function  of  a  as  well  as  of  0. 

The  primary  solutions  (B)    are   not  the  only  primary  solutions  of 

*  Math.  Ann.  vol.  LVII,  p.  333  (1903);  Whittaker  and  Watson,  Modern  Analysis,  ch.  xvin. 


Primary  Solutions  97 

Laplace's  equation,  for  it  was  shown  by  Jacobi*  that  if  9  is  defined  by  the 
equation  0  -  #  (9)  +  *,  (8)  +  *  (9),  ......  (F) 

where  £  (6),  rj  (0)  and  £  (0)  are  functions  connected  by  the  relation 

[f  W]2  +  h  (0)]2  +  K  Wl2  =  o, 

then  F  =  .F  (8)  is  a  solution  of  Laplace's  equation. 
This  is  easily  verified  because  iff 

M=l-x?  (d)  -  yr,'  (d)  -  z?  (6), 


we  have 


These  equations  give 


M     ~      '  [*"  (•>  +  OT"  w  +  *r  (»n  -  f  <»>    - 


and  so  V20  -  0,     V2  {F  (0)}  -  0. 

This  theorem  is  easily  generalised.  If  ca  (a),  c2  (a),  ...  cw  (a)  are  functions 
connected  by  the  identical  relation  (D)  the  quantity  6  defined  by  the 
equation  &  =  ^  ^  +  ^^  (&)  +  _  ^^  ((?)  (Q) 

is  such  that  i/  =  ^  (0)  is  a  primary  solution  of  equation  (A). 

Since  v  =  du/dxl  is  also  a  solution  of  the  same  differential  equation  it 
follows  that  if  G  (6)  is  an  arbitrary  function  and 

M  =  1  -  *lC/  (6)  -  x2c2'  (6)  -  ...  xmcm'  (0) 
tne  expression  v  =  M~1G  (0) 

is  a  second  solution  of  the  differential  equation.  The  reader  who  is  familiar 
with  the  principles  of  contour  integration  will  observe  that  this  solution 
may  be  expressed  as  a  contour  integral 

If  O  (a)  da 

V  — — i L 

2-rri ) c  a  -  X&  (a)  -  X2c2  (a)  -  ...  xmcm  (a) ' 

where  C  is  a  closed  contour  enclosing  that  particular  root  of  equation  (G) 
which  is  used  as  the  argument  of  the  function  G  (0). 

It  is  easy  to  verify  that  the  contour  integral  is  a  solution  of  the 
differential  equation  because  the  integrand  is  a  primary  solution  for  all 
values  of  the  parameter  a  and  has  been  generalised  by  the  method  already 
suggested. 

*  Journal  fiir  Math.  vol.  xxxvi,  p.  113  (1848);  Werke,  vol.  IT,  p.  208. 
f  We  use  primes  to  denote  differentiations  with  respect  to  0. 


98  The  Classical  Equations 

In  this  method  of  generalisation  by  integration  with  respect  to  a  para- 
meter the  limits  of  integration  are  generally  taken  to  be  constants  or  the 
path  of  integration  is  taken  to  be  a  closed  contour  in  the  complex  plane. 
It  is  possible,  however,  to  still  obtain  a  solution  of  the  differential  equation 
when  the  limits  of  integration  are  functions  of  the  independent  variables 
of  type  0.  Thus  the  integral 


re 
V  =       W  (z  +  ix  cos  a  +  iy  sin  a,  a)  da 

J  o 

V2F  =  0  when  6  is  < 
(a)   _  77  (a)  _  £  (a) 


satisfies  Laplace's  equation  V2F  =  0  when  0  is  defined  by  an  equation  of 
type  (F)  where 


icosa     ism  a        1 

When  the  equation  (A)  possesses  primary  solutions  of  type  u  =  F  (6) 
and  no  primary  solutions  of  type  u  =  F  (0,  <f>)  it  will  be  said  to  be  of  the 
first  grade.  When  it  possesses  primary  solutions  of  type  u  =  F  (0,  <f>)  and 
no  primary  solutions  of  type  u  =  F  (0,  <f>,  if/)  it  will  be  said  to  be  of  the 
second  grade  and  so  on. 

The  equation  d2u/dxdy  =  0  is  evidently  of  the  first  grade  because  the 
general  solution  is  u  =  F  (x)  -f  G  (y),  where  F  and  G  are  arbitrary 
functions.  The  primary  solutions  are  in  this  case  F  (x)  and  G  (y). 

Laplace's  equation  V2  (u)  =  0  is  also  of  the  first  grade  but  the  equation 

du     Su     du  _ 
dx+dy+dz** 
is  of  the  second  grade  because  the  general  solution  is  of  type 

u  =  F  (y  -  z,  z  -  x). 
There  is,  of  course,  a  primary  solution  of  type 

u=  F  (y-  z,z-  x,x-  y), 

where  F  (0,  </>,  0)  is  an  arbitrary  function  of  the  three  arguments  0,  <f>y  ^r, 
but  these  arguments  are  not  linearly  independent  ;  indeed,  since 

0  +  <£  -f  ^  =  0, 

a  function  of  0,  <f>  and  ifi,  is  also  a  function  of  0  and  <f>.    In  the  foregoing 
definition  of  the  grade  of  the  equation  it  must  be  understood,  then,  that 
the  parameters  0,  <f>,  $,  etc.,  are  supposed  to  be  functionally  independent. 
The  differential  equation 


has  not  usually  a  grade  higher  than  one.   If,  in  particular,  an  attempt  is 
made  to  find  a  solution  of  type 


where  0  =  x^  +  x2&  -f  z3£3  -f  a?4f4, 

<f>  =  #!*?!  +  #2*72  +  *3>?3  + 


Primary  Solution  of  the  Wave-Equation  99 

it  is  found  that  a  number  of  equations  must  be  satisfied.  These  equations 
imply  that 

where  a  and  b  are  arbitrary  parameters  and  this  means  that  all  points  of 
the  line 

lie  on  the  surface  whose  equation  is 

/(r      r      Y      v  \  —  0 
V^i>  ^2)  ^3?  •*'4;  ~~  u« 

When  /  (#! ,  x2 ,  x3 ,  x4)  has  a  linear  factor  of  the  first  degree  or  is  itself 
of  the  first  degree  the  equation  (A)  is  of  grade  3.  In  particular  the  equation 

possesses  the  general  solution 

Tfl     /  \ 

and  so  is  of  grade  3.  An  equation  with  m  independent  variables  which,  by 
a  simple  change  of  variables,  can  be  written  in  the  form 

a    /  a    a        a 


is  said  to  be  reducible.  Such  an  equation  is  evidently  of  grade  m  —  1.  It 
is  likely  that  whenever  the  number  of  independent  variables  is  m  and  the 
grade  m  —  1  the  equation  is  reducible.  The  wave-equation 


is  of  grade  2  because  there  is  a  primary  solution  of  type 

u  =  F(0,  <£), 

where        n  .... 

0  —  x  cos  a  4-  i/  sin  a  -f  tz,     q>  =  x  sin  a.  —  y  cos  a  4- 

This  solution  may  be  generalised  so  as  to  give  a  solution 

u=  !  'F  (0,<f>,  a)  da, 

Jo 

analogous  to  Whittaker's  solution  of  Laplace's  equation. 


Theequations 


,,_ 

c    St  ~    ' 


_. 
~dx       S^c  St 


7-2 


100  The  Classical  Equations 

which  may  be  written  in  the  abbreviated  form  * 


and  which  give  the  simple  equations  of  Maxwell 


curl  17  =-  -  3j^,     div#  =  0, 

C    vt 


when  the  vector  Q  is  replaced  by  H  +  iE,  where  E  and  //  are  real,  may  be 
satisfied  by  writing 


where  q  (a)  is  a  vector  with  components  cos  «,  sin  a,  i,  respectively  and 
F  (0,  <f>,  a)  is  an  arbitrary  function  of  its  arguments. 

EXAMPLES 

1.    Let  £,  v),  £,  T  he  functions  of  a,  /9,  y  connected  by  the  relation 

e  +  ^  +  j2  +  r2  =  i, 

and  let  X  =--  rx  -  fy  +  rjz  —  ft  +  u,     Z  =  —  yx  +  (y  +  TZ  —  &  +  w, 

Y  =  &  +  ry-  fz-iit  +  v,     T  =  (x  +  ijy  +  ^2  f-  T<  +  «. 
Prove  that  if  the  integration  extends  over  a  suitable  fixed  region  the  definite  integral 

V 

satisfies  the  differential  equation 


2.   If  V  =  F(A,B,C,D,E) 

is  a  solution  of  the  equation 

a2F    a2F    a2F    a2F_a^F 
a^2  +  dB*  +  BC*  +  en2  ~  BE* 

when  considered  as  a  function  of  A,  B,  C,  D  and  E;  then,  when 

A  =»  2  (xs  -f  yw  —  zv  —  tu),         G  «  2  (25  +  xv  —  yu  —  tw), 
B  =  2  (2/5  4-  zu  —  xw  —  tv),         D  =  2  (^  -f  xu  +  yv  -f  zw), 

#  =  a;2  +  y2  4-  z2  -f  ^2  -f  w2  +  t>2  +  ^2  +  *2, 
the  function  F  is  a  solution  of 

a27    a2F    a2F    a2F  =  a2F    a2F    &y    a2F 
a^2  +  at/2  +  dz2  +  a*2    a^2  4"  a^2  +  a^2  +  a? 

when  considered  as  a  function  of  x,  y,  z,  t,  u,  v,  w,  s. 

*  We  use  the  symbol  Q  to  denote  the  vector  with  components  Qx,  Qy,  Qt  respectively.  This 
abbreviated  form  is  due  to  H.  Weber  and  L.  Silberstein. 


Characteristics  101 

§"'1-92.  The  partial  differential  equation  of  the  characteristics.  It  is  easily 
seen  that  when  0  =  c^  -f  C2x2  +  ...  cmxm  and  cl9  c2,  ...  cm  are  constants 
satisfying  the  equation  /  (cl9  c2,  ...  cm)  =  0,  the  function  F  (0)  =  u  is  not 
only  a  primary  solution  of  the  equation  /  (A ,  Z>2,  ...  Z>m)  w  =  0  but  it  is 
also  a  solution  of  the  equation 

/(Aw,  A*,  -  AH")  =  O.  (A) 

This  partial  differential  equation  of  the  first  order  is  usually  called  the 
partial  differential  equation  of  the  characteristics  of  the  equation 

/(A,A>..-An)*  =  o.  (B) 

In  particular,  the  quantity  u  =  0  is  a  solution  of  this  differential 
equation  and  the  locus  0  (xl9  x2,  ...  xm)  =  constant  is  a  characteristic  or 
characteristic  locus  of  the  partial  differential  equation. 

A  characteristic  locus  can  generally  be  distinguished  from  other  loci  of 
type  (f>  (xl9x29  ...  xm)  =  constant  by  the  property  that  it  is  a  locus  of 
"singularities"  or  "discontinuities"  of  some  solution  of  the  differential 
equation.  If  we  adopt  this  definition  of  a  characteristic  locus  0  =  constant 
it  is  clear  that  0  =  constant  is  a  characteristic  locus  whenever  there  is  a 
solution  of  the  equation  which  involves  in  some  explicit  manner  an 
arbitrary  function  F(0)9  for  the  function  F(0)  can  be  given  a  form  which 
will  make  the  solution  discontinuous  on  the  characteristic  locus. 

Thus  the  quantity  u  =  e~lle  is  a  solution  of  the  differential  equation  (B) 
when  0  =£  0  and  is  discontinuous  at  each  point  of  the  characteristic  locus 
0  —  0.  It  should  be  observed  that  this  function  and  all  its  derivatives  on 
the  side  0  >  0  of  the  locus  0—0  are  zero  for  0=0.  The  function  u  =  e~llb* 
possesses  a  similar  property  and  the  additional  one  that  the  derivatives 
on  the  side  0  <  0  of  the  locus  0=0  are  also  zero.  From  these  remarks  it 
is  evident  that  if  there  is  a  solution  of  the  partial  differential  equation  (B) 
which  satisfies  the  condition  that  u  and  its  derivatives  up  to  order  n  —  1 
have  assigned  values  on  the  locus  <f>  (xly  x2,  ...  xm)  =  constant  and  so  gives 
the  solution  of  the  problem  of  Cauchy  for  the  equation,  this  solution  is  not 
unique  when  <f>  =  0  because  a  second  solution  may  be  obtained  by  adding 
to  the  former  one  a  solution  such  as  e~1/62  which  vanishes  and  has  zero 
derivatives  at  all  points  of  the  locus.  This  property  of  a  lack  of  uniqueness 
of  the  solution  of  the  Cauchy  problem  for  the  locus  0  (xl9  x29  ...  xm)  =  0  is 
the  one  which  is  usually  used  to  define  the  characteristic  loci  of  a  partial 
differential  equation  and  can  be  used  in  the  case  when  the  equation  does 
not  possess  primary  solutions.  Since,  however,  we  are  dealing  at  present 
with  equations  having  primary  solutions  the  simpler  definition  of  0  as  the 
argument  of  a  primary  solution  or  other  arbitrary  function  occurring  in  a 
solution  will  serve  the  purpose  quite  well. 

An  equation  with  a  solution  involving  an  arbitrary  function  explicitly 
(not  under  the  sign  of  integration)  will  be  called  a  basic  equation. 


102  The  Classical  Equations 

Let  us  now  write  p1  =  D^u,  p2  =  D2u,  ...  so  that  the  partial  differential 
equation  for  the  characteristics  may  be  written  in  the  form 

f(Pi>P2>  ...Pm)  =  0- 
The  curves  defined  by  the  differential  equations 

dxl     dx2  dxm  r 

lr==¥=='"T  ...... 

dpl     dp2  dpm 

are  called  the  bicharacteristics  *  of  the  equation  ;  they  are  the  character- 
istics of  the  equation  (A)  according  to  the  theory  of  partial  differential 
equations  of  the  first  order. 

When  pl  ,  p2  ,  .  .  .  are  eliminated  from  these  equations  it  is  found  that 

F  (dxl9dx2,  ...dxm)  =  0, 

where  F  (xl  ,  x2  ,  .  .  .  xm)  =  0  is  the  equation  reciprocal  to/  (pl  ,  p2  ,  ...  pm)  =  0 
in  the  sense  of  the  theory  of  reciprocal  polars. 

In  mathematical  physics  the  loci  of  type  u  =  constant,  where  u 
satisfies  the  equation  (A),  frequently  admit  of  an  interesting  interpretation 
as  wave-surfaces.  The  curves  given  by  the  equations  (C)  associated  with  the 
function  u  are  interpreted  as  the  rays  associated  with  the  system  of  wave- 
surfaces. 

In  the  particular  case  when  the  partial  differential  equation  of  the 
characteristics  is 


d6     36        36        36         36 

#-Si  +  u*+vt9  +  W& 
and  u,  v  and  w  are  constants  representing  the  velocity  of  a  medium  and 
V  is  another  constant  representing  the  velocity  of  propagation  of  waves 
in  the  medium,  the  differential  equations  of  the  bicharacteristics  are 

dx  __          dy          __  dz  __  dt 

~H5    TTa0=='"1d0    ITs^"""^    ^Tso^Je 
u  j*  -  v  V    v  -j*  -  v  V    w  ^  -  v  5-    -J, 

dt  ox        dt  dy         dt    4        oz     dt 

and  the  equation  obtained  by  eliminating  x-  -  ,  ^-  ,  ^-  ,  -^  is 


(dx  -  udt)*  +  (dy  -  vdt)2  +  (dz  -  wdt)*  = 

This  result  is  of  considerable  interest  in  the  theory  of  sound  and  may  be 
extended  so  as  to  be  applicable  to  the  case  in  which  u,  v,  w  and  V  are 
functions  of  x,  y,  z  and  t. 

It  may  be  remarked  that  if  we  have  a  solution  of  (D)  in  the  form  of  a 

complete  integral  .  ,  m 

r  &  0  =  t  -  T  -  g  (x,  y,  z,  a,  j3), 

*  See  J.  Hadamard's  Propagation  des  Ondes.  The  theory  is  illustrated  by  the  analysis  of  §  19. 


Bicharacteristics  and  Rays  103 

in  which  r,  a  and  j3  are  arbitrary  constants,  the  rays  may  be  obtained  by 
combining  the  foregoing  equation  with  the  equations 

39  _  A      ^  -  o 
a«-U'     3)8  ~"U' 

The  characteristics  of  a  set  of  linear  equations  of  the  first  order  may  be 
defined  to  be  the  characteristics  of  the  partial  differential  equation  obtained 
by  eliminating  all  the  dependent  variables  except  one.  The  relation  of  the 
primary  solutions  of  this  equation  to  the  dependent  variables  in  the  set  of 
equations  of  the  first  order  is  a  question  of  some  interest  which  will  now  be 
examined. 

Let  us  first  consider  the  equations 

du  __dv       du  _  dv 

dx~~dy'     dy^fa'  ......  W 

which  lead  to  the  equation 


In  this  case  the  quantity  w  =  u  +  v  satisfies  a  linear  equation  of  the 

first  order  ~        ~ 

cw  _  dw 

fa^dy' 

and  this  equation  possesses  the  primary  solution  w  =  F  (x  -f  y)  which  is 
also  a  primary  solution  of  the  equation  (F). 

Similarly  the  quantity  z  =  u  —  v  satisfies  the  equation 


which  possesses  a  primary  solution  z  =  G  (x  —  y)  which  is  also  a  primary 
solution  of  the  equation  (F). 

To  generalise  this  result  we  consider  a  set  of  m  linear  partial  differential 
equations  of  the  first  order, 

L^UI  +  Ll2u2  -f  ...  Llmum  -  0,  j 
L21U}  -f  L22uz  -f  ...  L2mum  =  0,  1 

LmlUi  +  Lm2u2  +  ...  Lmmum  =  Oj 
where  Lvq  denotes  a  linear  operator  of  type 

(P>  V>  l)  A  +  (P>  V>  2)  D2  +  ...  (p,  q,  m)  Dm, 

where  the  coefficients  (p,  q,  r)  are  constants. 

Multiplying  these  equations  by  coefficients  bl9  b2)  ...  bm  respectively, 
the  resulting  equation  is  of  the  form 

L  (a^  -f  a2u2  +  ...  amum)  =  0  ......  (H) 


104  The  Classical  Equations 

if  the  constants  blt  b2,  ...  6m;  al9  a2,  ...  am  are  of  such  a  nature  that 


4- 


a) 


and  the  operator  L  is  of  the  form 


where  the  operator  coefficients  Zl5  £2,  ...  /w  are  constants  to  be  determined. 
Equating  the  coefficients  of  the  operator  D  in  the  identities  (I)  we 

obtain  _,,    ,  v  7 

S6p(p,  ?;r)  =  aaJf. 

This  equation  indicates  that  if  z1)z2,  ...  zm;  yl  ,  t/2  >  -  •  •  2/m  are  arbitrary 
quantities,  the  bilinear  form 


can  be  resolved  into  linear  factors 


When  the  coefficients  bly  ...  bm  can  be  chosen  so  that  the  bilinear  form 
breaks  up  in  this  way  the  two  factors  will  give  the  required  coefficients 
al9  ...  am;  119  ...  lm  and  the  partial  differential  equations  will  give  an  ex- 
pression for  (Zji/j  -f  ...  amum  which  may  be  called  a  primary  solution  of  the 
set  of  linear  partial  differential  equations  of  grade  m  ~  1.  When  such  a 
solution  exists  the  system  is  said  to  be  reducible.  The  problem  of  finding 
when  a  set  of  equations  is  reducible  is  thus  reduced  to  an  algebraic  problem. 

Now  let  f2  denote  the  determinant 


Anl       An2  •••  Anro 

and  let  An , ...  AlTO  denote  the  co-factors  of  the  constituents  Ln,  L12,  ...  Llm 
respectively.    If  we  write 

it  is  easily  seen  that  the  last  m  —  1  equations  of  the  set  are  all  formally 
satisfied,  and  since 

the  first  equation  is  formally  satisfied  if  v  is  a  solution  of  the  partial 
differential  equation 

which  is  of  order  m.    Since 

£}&!  =  flAuv  =  Au£lv  —  0, 

the  quantities  Ui,u2,  ...  um  are  all  solutions  of  the  same  partial  differential 
equation. 


=  0, 


Eeducibility  of  Equations  105 

It  should  be  noticed  that 

a^  +  ...  amum  =  (a1An  +  a2A12  +  ...  amAlm)  v, 
consequently 

L  (fl^Wj  +  ...  amum)  =  L  (a1An  -f  a2A12  +  ...  amAlm)  v. 

The  equation  £  (a^  +  ...  amum)  =  0  will  be  a  consequence  of  the  equation 
£lv  —  0  if  the  operator  Cl  breaks  up  into  two  factors  L  and 

(axAn  +  a2A12+  ...  amAlm) 

of  which  one,  L,  is  linear.  The  set  of  linear  equations  is  thus  reducible  when 
the  equation  £lv  =  0  is  reducible. 

It  is  clear  from  this  result  that  we  cannot  generally  expect  a  set  of  linear 
homogeneous  equations  of  type  (G)  to  possess  primary  solutions  of  grade 
m  —  1. 

The  equations  do,  however,  generally  possess  primary  solutions  of 
grade  1.  To  see  this  we  try 

*i=/i(0),     «2=/2(0),  -  *m=/*(0). 

Substituting  in  the  set  of  equations  we  obtain  the  set  of  linear  equations 
//  (0)  Ln0  +  /2'  (0)  L120  +  ...  fm'  (0)  Llm6  =  0, 
//'(»)  V  +/•'  (0)  L220  +  .../m'  (0)  L2m0  =  0, 


from  which  the  quantities//  (0),  f2  (0),  ...  fm'  (0)  may  be  eliminated.  The 

resulting  equation,  T    a      T    a        T     a 

LnU      L12V  ...  Llmu 

L210      L220  ...  L2m0 

.  Lml0    Lm20  ...  Lmm0 

is  no  other  than  the  partial  differential  equation  of  the  characteristics  of 
the  equation  £lu  =  0. 

§  1-93.  Primary  solutions  of  the  second  grade.  We  have  already  seen 
that  the  wave-equation  possesses  primary  solutions  of  type  F  (0y  <f>)  which 
may  be  called  primary  solutions  of  the  second  grade.  The  result  already 
obtained  may  be  generalised  by  saying  that  if  10,  m0,  n0,  p0,  119  ml9  nl9  pl 
are  quantities  independent  of  #,  t/,  z  and  t  and  connected  by  the  relations 

the  quantities 

are  such  that  the  function  u  =  F  (0,  <f>)  is  a  solution  of  Q2u  =  0. 


106  The  Classical  Equations 

This  result  may  be  generalised  still  further  by  making  the  coefficients 
Z0,  m0,  etc.,  functions  of  two  parameters  a,  r  and  forming  the  double  con- 
tour integral 

tt=_  Jiff  _  f(a,T)dadT  _ 
47T2JJ  (I0x 


_ 

(I0x  +  m0y  +  nQz  -  pQct  -  g0)  (^x  -f 

where/  (a,  T),  gr0  (a,  r),  ft  (a,  r)  are  arbitrary  functions  of  their  arguments. 

This  integral  will  generally  be  a  solution  of  the  wave-equation  and  the 

value  of  the  integral  which  is  suggested  by  the  theory  of  the  residues  of 

double  integrals  is  T  ,  .  .     n. 

*  u  =  J-*f(a,P), 

in  which  a,  /?  satisfy 

M«>j8)  =  0,        Ao(a,j8)  =  0,  ......  (K) 

where 

A!  (a,  r)  -  #/!  (a,  r)  +  yml  (a,  r)  +  zn^  (a,  r)  -  Ctp1  (a,  r)  -  ft  (a,  T), 
A0  (<r,  r)  =  xl0  (a,  r)  +  ymQ  (a,  r)  -f  2W0  (a,  r)  -  Ctp0  (a,  r)  -  gr0  (a,  r), 
and  J  is  the  value  when  cr  =  a,  r  =  ^8  of  the  Jacobian 


This  result,  which  may  be  extended  to  any  linear  equation  with  a  two- 
parameter  family  of  primary  solutions  of  the  second  grade,  will  now  -be 
verified  for  the  case  of  the  wave-equation.  It  should  be  remarked  that  the 
method  gives  us  a  solution  of  the  wave-equation  of  type 


where  y  is  a  particular  solution  of  the  wave-equation.  Such  a  solution  will 
be  called  a  primitive  solution  ;  it  is  easily  verified  that  the  parameters  a  and 
j3  occurring  in  a  primitive  solution  are  such  that  the  function  v  =  F  (a,  j8) 
is  a  solution  of  the  partial  differential  equation  of  the  characteristics 


Instead  of  considering  the  wave-equation  it  is  more  advantageous  to 
consider  the  set  of  partial  differential  equations  comprised  in  the  vector 
equations  .  «  ^ 

=0  ......  (M) 


and  to  look  for  a  primitive  solution  of  these  equations  of  type 


in  which  /  is  an  arbitrary  function  of  the  two  parameters  a  and  /?,  which 
are  certain  functions  of  x,  y,  z  and  t,  and  the  vector  q  is  a  particular  solution 
of  the  set  of  equations. 

Substituting  in  the  equations  (L)  we  find  that  since  /is  arbitrary  a  and 
]8  must  satisfy  the  equations  in  o>, 

cVco  x  q  =  —  iqda/dt,         <?.Vaj  =  0, 


Primitive  Solution  of  Maxwell's  Equations  107 

which  indicate  that 


^    3  («.£)_**  8  («,£) 
-Kd(y,z)~  c  3(x,t)' 

_     a  (a,  )8)     uc  9  (a.  )8) 
"3(2,*)      c8(y,  0' 

8  «»          *«  9  (<*. 


(N) 


where  K  is  some  multiplier.  To  solve  these  equations  we  take  a,  /?,  #,  y  as 
new  independent  variables  and  write  the  equation  connecting  a  and  ft  in 

the  form  «  ,     ~       JX      •  ^  /     o        ^ 

8  (g,  ft,  x,  t)  =  *  8  (a,  ft,  y,  g) 

3  (y,  z,  x,t)     cd  (x,  t,  y,  z)  ' 


_ 
(z,  x,  y,t)     c  d  (y,  t,  z,  x)  ' 


_  ^ 

9  (a:,  y,  g,  «)      c  3  (z,  t,  x,  y)  ' 

Now  multiply  each  of  the  Jacobians  by  ~-?J-%-*—  \  and  make  use  of 
*J  J  a(a,j3,»,y) 

the  multiplication  theorem  for  Jacobians.  We  then  obtain  a  set  of  equations 
similar  to  the  above  but  with  3  (a,  /J,  x,  y)  in  each  denominator.  The  new 
equations  reduce  to  the  form 

dt  __       i  dz        dt  _  i  dz        d  (z,  t)  _  i 
dy          cdx'     dx  ~~  c  dy  '     8  (x,  y)     c  ' 

The  first  two  of  these  equations  are  analogous  to  the  equations  con- 
necting conjugate  functions  t  and  ig/c,  consequently  we  may  write 

z-ct  =  &[x+  iy,a,  /?], 
z  +  ct  =  g  [x  —  4y*  a,  j8]  . 

Substituting  in  the  third  equation,  we  find  that 

$'£'  =  -  1, 

where  in  each  case  the  prime  denotes  a  derivative  with  respect  to  the  first 
argument.  Evidently  &'  must  be  independent  of  x  -f  iy  and  <£'  inde- 
pendent of  x  —  iy.  The  general  solution  is  thus  determined  by  equations 
of  the  form  8  _  d  _  ^  (a,  0  +  (jr  +  iy)  d  («,  ft, 

Z  +  ct  =  4,(a,p)-(x-  iy)  [0  (a,  /3)]-1, 

where  0,  <j>,  ifi  are  arbitrary  functions  of  a  and  /9  which  are  continuous  (Z),  1) 
in  some  domain  of  the  complex  variables  a  and  ft. 

For  some  purposes  it  is  more  convenient  to  write  the  equations  in  the 

equivalent  form  ,       ,  /     rt.      /        .  v  n  ,     /,, 

H  z  -  ct  =  <£  (a,  j8)  +  (x  +  ty)  6  (a,  /5), 

0  (a,  ft  (z  +  c<)  =  X  («,  0)  -  (*  -  *»• 


108  The  Classical  Equations 

These  equations  are  easily  seen  to  be  of  the  type  (K)  and  may  indeed 
be  regarded  as  a  canonical  form  of  (K).  When  the  expressions  for  q  are 
substituted  in  the  equations  (M)  it  is  easily  seen  that  K  is  a  function 
of  a  and  ft.  Since  Q  already  contains  an  arbitrary  function  of  a  and  ft  we 
may  without  loss  of  generality  take  K  =  1. 

A  case  of  particular  interest  arises  when 

<£  =  £  (a)  -  cr  (a)  -  [f  (a)  +  iy  (a)]  0, 
0  =  £  (a)  +  cr  (a)  +  [£  (a)  -  irj  (a)]  6~\ 

where  f  (a),  77  (a),  £  («)  and  r  («)  are  real  arbitrary  functions  of  a  which  are 
continuous  (D,  2).    We  then  have 

^TI^"T[^^i^)j  ^  ~  r~~  Ti^+7ir-~T(a)] ' 

and  so  a  is  defined  by  the  equation 

We  may  without  loss  of  much  generality  take  r  (a)  =  a  and  use  r  as 
variable  in  place  of  a.  Let  us  now  regard  £  (T),  77  (T),  £  (T)  as  the  co- 
ordinates of  a  point  S  moving  with  velocity  v  which  is  a  function  of  r. 
For  the  sake  of  simplicity  we  shall  suppose  that  for  each  value  of  r  we 
have  the  inequality  v2  <  c2,  which  means  that  the  velocity  of  S  is  always 
less  than  the  velocity  of  light.  We  shall  further  introduce  the  inequality 
T  <  t.  This  is  done  to  make  the  value  of  r  associated  with  a  given  space- 
time  point  (x,  y,  z,  t)  unique*. 

To  prove  that  it  is  unique  we  describe  a  sphere  of  radius  c  (t  —  r)  with 
its  centre  at  the  point  occupied  by  S  at  the  instant  r.  As  r  varies  we 
obtain  a  family  of  spheres  ranging  from  the  point  sphere  corresponding 
to  r  =  t  to  a  sphere  of  infinite  radius  corresponding  to  r  =  —  oo. 

Now,  since  v*  <  c2  it  is  easily  seen  that  no  two  spheres  intersect.  Each 
sphere  is,  in  fact,  completely  surrounded  by  all  the  spheres  that  correspond 
to  earlier  times  r.  There  is  consequently  only  one  sphere  through  each 
point  of  space  and  so  the  value  of  r  corresponding  to  (x,  y,  z,  t)  is  unique. 
The  corresponding  position  of  8  may  be  called  the  effective  position  of  S 
relative  to  (x,  y,  z,  t). 

In  calculating  the  Jacobians  r  may  be  treated  as  constant  in  the 
differentiations  of  ft.  Now 


*  Proofs  of  this  theorem  have  been  given  by  A.  Li6nard,  L'&lairage  tiectrique,  t.  xvi,  pp.  5,  i  ^, 
106  (1898);  A.  W.  Con  way,  Proc.  London  Math.  Soc.  (2),  vol.  I  (1903);  G.  A.  Schott,  Electromagnetic 
Radiation  (Cambridge,  1912). 


Primitive  Solution  of  the  Wave-Equation  109 

where 

M=[x-£  (r)]  £'(T)  +  (y  ~  V  (T)]  VW  +  [Z  -  I  (r)]  £'(T)  -  fa  ('  -  T), 
and  primes  denote  derivatives  with  respect  to  T.    We  thus  find  that 

'§(y,  2)=~2J/(1~^)> 

a  («,  0)       »/s 


a(3,y)  JIT 

The  ratios  of  the  Jacobians  thus  depend  only  on  ft  and  we  have  the  general 
result  that  the  function  M-I  f  /     R\ 

is  a  solution  of  the  wave-equation. 

When  the  point  (£,  T?,  £)  is  stationary  and  at  the  origin  of  co-ordinates 
this  result  tells  us  that  if,/  is  an  arbitrary  function  which  is  continuous 
(/),  2)  in  some  domain  of  the  variables  a,  ft  and  if  r2  =  x*  +  ?/2  -I-  z2  the 

function  , 

z  -  r 

*  +  ?y 

is  a  solution  of  the  wave-equation.  There  is  a  corresponding  primitive 
solution  of  type 


obtained  by  changing  the  sign  of  t  and  using  another  arbitrary  function. 

In  the  case  of  the  wave-function  M~lf  (a,  /J)  the  parameter  a  may  be 
called  a  phase-parameter  because  it  determines  the  phase  of  a  disturbance 
which  reaches  the  point  (x,  y,  z)  at  time  t  when  the  function  /  is  periodic 
in  a.  The  parameter  ft  is  on  the  other  hand  a  ray-parameter  because  a 
given  complex  value  of  /?  determines  the  direction  of  a  ray  when  a  is  given. 

It  is  easily  deduced  from  the  equations  (N)  that  «  and  /3  satisfy  the 
differential  equation  of  the  characteristics 


, 

and  that  + 


It  follows  that  the  quantity  v  =  F  (a,  ft)  is  also  a  solution  of  (L). 

An  interesting  property  of  this  equation  (0)  is  that  if  a  is  any  solution 
and  we  depart  from  the  space-time  point  (x,  y,  z,  t)  in  a  direction  and 
velocity  defined  by  the  equations 


dx     dy      dz  -r]«  sav 

_  =  _  =  —-  =  —  -  --  as,  say, 

da      ca      da  da 

dx     dy      dz  dt 


110  The  Classical  Equations 

then  a  and  its  first  derivatives  are  unaltered  in  value  as  we  follow  the 
moving  point.   We  have  in  fact 

,        da  7        da  ,     ,  da  ,       da  7j 
da=^dx+*ray+~  dz  +  ^  at 

ox  oy   *      oz          ot 


2  2 


3%    ,         32a    ,     r     32« 

+  3—3"  ^  + 


[3a  32 
3i  fa 


32«     3«    32«       3a  32a       1  da  , 

3~z  3iaz  "  c2 


-  0. 
Also,  if  a  and  /J  are  connected  by  an  equation  of  type  (P), 


3a  36      1  3a  3B\  7 

,         .i_  „  —   ^_     .  "  )  /7o 

dz  dz      c2  dt  dtj 
=  0. 

The  equations  (0  and  P)  thus  indicate  that  the  path  of  the  particle 
which  moves  in  accordance  with  these  equations  is  a  straight  line  described 
with  uniform  velocity  c  and  is,  moreover,  a  ray  for  which  j3  is  constant. 

§  1*94.  Primitive  solutions  of  Laplace's  equation.  As  a  particular  case 
of  the  above  theorem  we  have  the  result  that  the  function 


r    \x  4-  iy/ 

is  a  primitive  solution  of  Laplace's  equation.  This  is  not  the  only  type  of 
primitive  solution,  for  the  following  theorem  has  been  proved*. 

In  order  that  Laplace's  equation  may  be  satisfied  by  an  expression  of 
the  form  V  =  yf  (0),  in  which  the  function  /  is  arbitrary,  the  quantity  6 
must  either  be  defined  by  an  equation  of  the  form 

[^  -  £  (6)1*  +(y~r,  (*)]»  +  [z  -  C  (0)?  =  o, 
or  by  an  equation  of  the  form 

xl  (6)  +  ym  (6)  +  zn  (6)  =  p  (6), 
where  /,  m,  n  are  either  constants  or  functions  of  6  connected  by  the  re- 

lation  P  +  m»  +  n«-0. 

The  most  general  value  of  y  is  in  each  case  of  the  form 
y  =  no  (6)  +  y*b  (6), 

*  See  my  Differential  Equations,  p.  202. 


Primitive  Solutions  of  Laplace's  Equation  111 

where  yx  and  y2  are  particular  values  of  y,  whose  ratio  is  not  simply  a 
function  of  9.   In  the  first  case  we  may  take 

n  =  w-t,    72  =  %-*, 

where         u;  =  [a;  -  £  (0)]  A  (0)  +  [y  -  rj  (6)]  ^  (0)  +  [2  -  £  (0)  ]  „  (0), 
t*i  =  [s  -  £  (#)]  \  (0)  +  [y  -  rj  (6)]  ^  (6)  +  [z  -  £  (6)]  v,  (6), 

and  A,  ^,  v,  Ax,  /zx,  vl  are  two  independent  sets  of  three  functions  of  6  which 
satisfy  relations  of  type 

A2  +  ^  +  v2  =  0, 

A  (0)  r  (0)  +  p,  (0)  r)'  (6)  +  v  (6)  £'  (0)  =  0. 
In  the  second  case  we  may  take  yl  ~  1  and  define  y2  by  the  equation 

yri  =  ^  (fl)  +  ym'  (9)  +  zri  (9)  -  p'  (9). 
If  in  the  first  theorem  we  choose  £  =  0,  rj  —  i9,  £  ~  0,  we  have 

r2 

9=-  -----  .--,     A-f  iu=  0,     v  =  0, 

x  +  ly  ^ 

iv  =  x  +  iy, 
and  the  theorem  tells  us  that  the  function 


is  a  primitive  solution  of  Laplace's  equation.    If  we  write  x  +  iy  =  t, 
x  —  iy  =  4:8  this  theorem  tells  us  that  the  function 

F=r*/(4*  +  z2/*) 
is  a  primitive  solution  of  the  equation 

d*V 


_ 

dz*  ~ 

§  1-95.  Fundamental  solutions*.  The  equations  with  primary  and 
primitive  solutions  have  been  called  basic  because  it  is  believed  that 
solutions  of  a  differential  equation  with  the  same  characteristics  as  a  basic 
equation  can  be  derived  from  solutions  of  the  basic  equation  by  some 
process  of  integration  or  summation  in  which  singularities  of  these  solutions 
of  the  basic  equation  fill  the  whole  of  the  domain  under  consideration. 

This  point  will  be  illustrated  by  a  consideration  of  Laplace's  equation 
as  our  basic  equation. 

We  have  seen  that  there  is  a  primitive  solution  of  type 


r  *  \x  + 
By  a  suitable  choice  of  the  function  /  we  obtain  a  primitive  solution 

*  These  are  also  called  elementary  solutions.  See  Hadamard,  Propagation  des  Ondes. 


112  The  Classical  Equations 

with  singularities  at  isolated  points  and  along  isolated  straight  lines  issuing 
from  isolated  singular  points.  The  particular  solution 

F=  1/r 

has  the  single  isolated  point  singularity  x  =  0,  y  =  0,  z  =  0.  Let  us  take 
this  particular  solution  as  the  starting-point  and  generalise  it  by  forming 
a  volume  integral 


over  a  portion  of  space  which  we  shall  call  the  domain  &). 

When  the  point  (x,  y,  z)  is  in  the  domain  I/1  this  integral  is  riot  a  solution 
of  Laplace's  equation  but  is  generally  a  solution  of  the  equation 

V2  V  +  *vF(x,  y,  3)  =  0,  ......  (B) 

provided  suitable  limitations  are  imposed  upon  the  function  F. 

Now  the  function  F  is  at  our  disposal  and  in  most  cases  it  can  be  chosen 
so  as  to  represent  the  terms  which  make  the  given  differential  equation 
differ  from  the  basic  equation  of  Laplace.  It  is  true  that  this  choice  of  F 
does  not  give  us  a  formula  for  the  solution  of  the  given  equation  but  gives 
us  instead  an  integro-differential  equation  for  the  determination  of  the 
solution.  Yet  the  point  is  that  when  this  equation  has  been  solved  the 
desired  solution  is  expressed  by  means  of  the  formula  (A)  in  terms  of 
primitive  solutions  of  the  basic  equation. 

A  solution  of  the  basic  equation  which  gives  by  means  of  an  integral  a 
solution  of  the  corresponding  equation,  such  as  (B),  in  which  the  additional 
term  is  an  arbitrary  function  of  the  independent  variables,  is  called  a 
fundamental  solution.  Rules  for  finding  fundamental  solutions  have  been 
given  by  Fredholm  and  Zeilon.  In  some  cases  the  solution  which  is  called 
fundamental  seems  to  be  unique  and  the  theory  is  simple.  In  other  cases 
difficulties  arise.  In  any  case  much  depends  upon  the  domain  W  and  the 
supplementary  conditions  that  are  imposed  upon  the  solution. 

When  the  basic  equation  is  the  wave-equation  the  question  of  a  funda- 
mental solution  is  particularly  interesting.  There  are,  indeed,  two  solutions, 


J7     1  I     1         1 

and  F=   -      .-}- 


2r  [r-ct  '  r  +  ct]      r*  -  cW 
which  may  be  regarded  as  natural  generalisations  of  the  fundamental 
solution  1/r  of  Laplace's  equation.  The  former  seems  to  be  the  most  useful 
as  is  shown  by  a  famous  theorem  due  to  Kirchhoff . 

In  the  case  of  the  equation  of  the  conduction  of  heat  the  solution  which 
is  regarded  as  fundamental  is 


Fundamental  Solutions  113 

when  the  equation  is  taken  in  the  form 


_    X* 

and  is  V  =  t~^  e    *** 

when  the  equation  is  taken  in  the  simpler  form 


The  equation  of  heat  conduction  is  not  a  basic  equation  but  may  be 
transformed  into  a  basic  equation  by  the  introduction  of  an  auxiliary 
variable  in  a  manner  already  mentioned.  Thus  the  basic  equation  derived 
from  97 


w 

is  =-3-  =  -s-9-  ,     W  = 

3s  3J      dx2 

and  this  equation  possesses  the  primitive  solution 


of  which  TF  =  2~i  exp    —  - 

r    \_K         4:Ktj 

is  a  special  case. 

The  theory  of  fundamental  solutions  is  evidently  closely  connected 
with  the  theory  of  primitive  solutions  but  some  principles  are  needed  to 
guide  us  in  the  choice  of  the  particular  primitive  solution  which  is  to  be 
regarded  as  fundamental.  The  necessary  principles  are  given  by  some 
general  theorems  relating  to  the  transformation  of  integrals  which  are 
forms  or  developments  of  the  well-known  theorems  of  Green  and  Gauss. 
These  theorems  will  be  discussed  in  Chapter  II.  An  entirely  different 
discussion  of  the  fundamental  solutions  of  partial  differential  equations 
with  constant  coefficients  has  been  given  recently  by  G.  Herglotz,  Leipzig  er 
Berichte,  vol.  LXXVIII,  pp.  93,  287  (1926)  with  references  to  the  literature. 

EXAMPLES 

1.   Prove  that  the  equation  -.    =  ^-j 

ot       dxr 

is  satisfied  by  the  two  definite  integrals 

V  -  4  f  °°  e-**  (cos  xs  -  sin  xs)  e'*t8'  da, 

/GO 

F=*/     v(a,t)v(x,8)ds, 


where  v  (x,  t) 

Show  also  that  the  two  integrals  represent  the  same  solution. 


114  The  Classical  Equations 

2.   Prove  that  this  solution  can  be  expanded  in  the  form 

V  -  F0  -  F,  +  F», 

where  F0  =  r (i) (4<F l [l  - 4,  fj  +  ^  (5)"  +  •••) 


3.  Show  also  that 

y  =  4  /     e~4<*4  cos  sx  cosh  sxds, 
7o 

Vl  —  4  I     e~*i8*  [sin  sx  cosh  sx  +  cos  sx  sinh  50;]  e&, 
Fo  ==  4  I     e~4<s4  sin  sx  sinh  so; .  ds. 

2       Jo 

4.  Prove  that  there  is  a  fourth  solution 

V3  -  x?1rl    oi  ~  7  i  T  "^"  n  i"\l  (  <  )    ~  "•    "  ^  /     e~4<s4  tsm  5a:  C08^  5a:  —  cos  sx  smn  5a:]  ^5- 

5.  If  V  (x,  t)  is  a  solution  of  the  equation 

3V  __  d*V 

~df  "  to*         *  ~  lj 
the  quantity 


is  generally  a  solution  of  the  set  of  equations 

In  particular,  if  s  =.  2  and  V  (x,  t)  is  the  function  v  (x,  t)  of  Ex.  1,  the  corresponding 
function  yn  (t)  is 


yn  (0 


This  may  be  called  the  fundamental  solution,  and  when  the  second  form  is  adopted 
s  may  have  any  positive  integral  value.  In  particular,  when  5  =  4,  this  function  is  de- 
rivable from  t^e  function  v  (x,  t)  of  Ex.  1,  p.  113. 


CHAPTER   II 

APPLICATIONS   OF   THE   INTEGRAL   THEOREMS 
OF   GAUSS   AND   STOKES 

.   In  the  following  investigations  much  use  will  be  made  of  the 
well-known  formulae 


......  (A) 

for  the  transformation  of  line  and  surface  integrals  into  surface  and  volume 
integrals  respectively.  In  these  equations  Z,  m,  n  are  the  direction  cosines 
of  the  normal  to  the  surface  element  dS,  the  normal  being  drawn  in  a 
direction  away  from  the  region  over  which  the  volume  integral  is  taken  or 
in  a  direction  which  is  associated  with  the  direction  of  integration  round 
the  closed  curve  C  by  the  right-handed  screw  rule. 

The  functions  u,  v,  w,  X,  Y,  Z  occurring  in  these  equations  will  be 
supposed  to  be  continuous  over  the  domains  under  consideration  and  to 
possess  continuous  first  derivatives  of  the  types  required*  .  The  equations 
may  be  given  various  vector  forms,  the  simplest  being  those  in  which 
u,  v,  w  are  regarded  as  the  components  of  a  vector  q  and  X,  Y,  Z  the 
components  of  a  vector  F.  The  equations  are  then 

I  q  .  ds  =  I  (curl  q)  .  dS, 

j    _   J 

J  F.  dS  =  [(div  F)  dr     (dr  =  dx  .  dy  .  dz), 

—^ 
where  ds  now  stands  for  a  vector  of  magnitude  ds  and  the  direction  of  the 

tangent  to  the  curve  C,  while  dS  represents  a  vector  of  magnitude  dS  and 
the  direction  of  the  outward-drawn  normal.  The  dot  is  used  to  indicate  a 
scalar  product  of  two  vectors.  Another  convenient  notation  is 

\qtds  =  I  (curlq)ndS, 

J  ~  ......  (C) 


/  FndS     f(divF)dr, 


*  See  for  instance  Goursat-Hedrick,  Mathematical  Analysis,  vol.  I,  pp.  262,  309.  Some  in- 
teresting remarks  relating  to  the  proofs  of  the  theorems  will  be  found  in  a  paper  by  J.  Carr,  Ph  il. 
Mag.  (7),  vol.  iv,  p.  449  (1927).  The  first  theorem  is  well  discussed  by  W.  H.  Young,  Proc.  London 
Math.  Soc.  (2),  vol.  xxiv,  p.  21  (1926);  and  by  O.  D.  Kellogg,  Foundations  of  Potential  Theory, 
Springer,  Berlin  (1929),  ch.  iv. 


8-2 


116    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

where  the  suffixes  t  and  n  are  used  to  denote  components  in  the  direction 
of  the  tangent  and  normal  respectively. 

If  we  write  Z  -  v,  Y  =  -  w,  X  =  0;  X  =  w,  Z  =  -  u,  Y  =  0;  7  =  ?/, 
X  =  —  v,  Z  =  0  in  succession  we  obtain  three  equations  which  may  be 
written  in  the  vector  form 


I  (qx  dS)  =  -  j(curlq)dr,  (D) 


where  the  symbol  x  is  used  to  denote  a  vector  product. 

Again,  if  we  write  successively  X  =  p,  Y  =  Z  =  0;  Y  =  p,  Z  =  X  =  0; 
Z  =  p,  X  =  Y  =  0,  we  obtain  three  equations  which  may  be  written  in 
the  vector  form  ^ 

......  (E) 


where  Vp  denotes  the  vector  with  components  ^  ,  ^  ^  ,  ~  respectively. 

§  2-12.  To  obtain  physical  interpretations  of  these  equations  we  shall 
first  of  all  regard  u,  v,  w  as  the  component  velocities  of  a  particle  of  fluid 
which  happens  to  be  at  the  point  (#,  y,  z)  at  time  t.  The  quantities  £,  77,  £ 
defined  by  the  equations 

.  _  dw     dv          _du     dw      ^  _  Sv      du 
*=dy~dz'     r]~dz~dx9     ^^dx^dy 

may  then  be  regarded  as  the  components  of  the  vorticity. 

The  line  integral  in  (A)  is  called  the  circulation  round  the  closed  curve 
C  and  the  theorem  tells  us  that  this  is  equal  to  the  surface  integral  of  the 
normal  component  of  the  vorticity.  When  there  is  a  velocity  potential 
</>  we  have  ~  ,  -  ,  ~  , 

dd>  O(h  o<p 

u  =  -f-  ,     v  =  -£-  ,     w  =  /- 
dx  dy  dz 

(in  vector  notation  q  —  Vcf>)  and  £  =  y  =  £  =  0,  the  circulation  round  a 
closed  curve  is  thenxzero  so  long  as  the  conditions  for  the  transformation 
of  the  line  integral  into  a  surface  integral  are  fulfilled.  The  circulation  is 

not  zero  when  .       ,       .  ,    ,  x 

</>  =  tan-1  (y/x), 

and  the  curve  C  is  a  simple  closed  curve  through  which  the  axis  of  z  passes 
once  without  any  intersection.  The  axis  of  z  is  then  a  line  of  singularities 
for  the  functions  u  and  v.  The  value  of  the  integral  is  27r,  for  ^  increases  by 
2n  in  one  circuit  round  the  axis  of  z.  The  velocity  potential 

</>  =  (r/27r)  tan-*  (y/x) 

may  be  regarded  as  that  of  a  simple  line  vortex  along  the  axis  of  z,  the 
strength  of  the  vortex  being  represented  by  the  quantity  F  which  is- 
supposed  to  be  constant.  F  represents  the  circulation  round  a  closed  curve 
which  goes  once  round  the  line  vortex. 


Equation  of  Continuity  1F7 

If  we  write  X  =  pu,  Y  =  pv,  Z  =  pw,  where  p  is  the  density  of  the  fluid, 
the  surface  integral  in  (A)  may  be  interpreted  as  the  rate  at  which  the 
mass  of  the  fluid  within  the  closed  surface  S  is  decreasing  on  account  of 
the  flow  across  the  surface  8.  If  fluid  is  neither  created  nor  destroyed 
within  the  surface  this  decrease  of  mass  is  also  represented  by 


The  two  expressions  are  equal  when  the  following  equation  is  satisfied  at 
each  place  (#,  y,  z)  and  at  each  time  t, 


This  is  the  equation  of  continuity  of  hydrodynamics.  There  is  a  similar 
equation  in  the  theory  of  electricity  when  p  is  interpreted  as  the  density 
of  electricity  and  u,  v,  w  as  the  component  velocities  of  the  electricity 
which  happens  to  be  at  the  point  (#,  ?/,  z)  at  time  t.  When  p  is  constant 
the  equation  of  continuity  takes  the  simple  form 

du     dv      div  __ 

dx  +  dy  +  dz  =  ° 

(in  vector  notation  div  q  =  0).  This  simple  form  may  be  used  also  when 
dp/dt  =  0,  where  d/dt  stands  for  the  hydrodynamical  operator 

d     a       a       a        a 

i4  =  ^  +  u  n    +  v  ~T  +  w  a~  » 
«£      cM          ra         cty          dz 

a  fluid  for  which  rfp/Y/£  =  0  is  said  to  be  incompressible. 

When  p  is  interpreted  as  fluid  pressure  the  equation  (E)  indicates  that 
as  far  as  the  components  of  the  total  force  are  concerned  the  effect  of  fluid 
pressure  on  a  surface  is  the  same  as  that  of  a  body  force  which  acts  at  the 
point  (x9  y,  z)  and  is  represented  in  magnitude  and  direction  by  the  vector 
—  Vp,  the  sign  being  negative  because  the  force  acts  inwards  and  not 
outwards  relative  to  each  surface  element.  Putting  q  =  pr  in  equation  (D), 
where  r  is  the  vector  with  components  x,  y,  z,  we  have  an  equation 

I  (r  x  pds)  =  —  I  (curler)  dr  =    (r  x  Vp)  dr, 

which  indicates  that  the  foregoing  distribution  of  body  force  gives  the 
same  moments  about  the  three  axes  of  co-ordinates  as  the  set  of  forces 
arising  from  the  pressures  on  the  surface  S.  The  body  forces  are  thus 
completely  equivalent  to  the  forces  arising  from  the  pressures  on  the 
surface  elements.  This  result  is  useful  for  the  formulation  of  the  equations 
of  hydrodynamics  which  are  usually  understood  to  mean  that  the  mass 
multiplied  by  the  acceleration  of  each  fluid  element  is  equal  to  the  total 
body  force.  If  in  addition  to  the  body  force  arising  from  the  pressure  there 
is  a  body  force  F  whose  components  per  unit  mass  are  JL  ,  7,  Z  for  a  particle 


118    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

which  is  at  (x,  y,  z)  at  time  t,  the  equations  of  hydrodynamics  may  be 
written  in  the  vector  form 


When  viscosity  and  turbulence  are  neglected  the  body  force  often  can  be 
derived  from  a  potential  fi  so  that  F  =  VQ.  The  hydrodynamical  equations 
then  take  the  simple  form  , 


which  implies  that  in  this  case  there  is  an  acceleration  potential  if  p  is  a 
constant  or  a  function  of  p.  When  in  addition  there  is  a  velocity  potential 
<f>  the  equations  may  be  written  in  the  form 


and  imply  that  f  dp~  +  ?J  +  |92  =  Q  +  /  (t), 

J     p          01 

where/  (t)  is  some  function  of  t.  This  may  be  regarded  as  an  equation  for 
the  pressure,  when  ti  =  0  it  indicates  that  the  pressure  is  low  where  the 
velocity  is  high. 

§  2-13.  The  equation  of  the  conduction  of  heat.  When  different  parts  of 
a  body  are  at  different  temperatures,  energy  in  the  form  of  heat  flows  from 
the  hotter  parts  to  the  colder  and  a  state  of  equilibrium  is  gradually 
established  in  which  the  temperature  is  uniformly  constant  throughout 
the  body,  if  the  different  parts  of  the  body  are  relatively  at  rest  and  do 
not  participate  in  an  unequal  manner  in  heat  exchanges  with  other  bodies. 
When,  however,  a  steady  supply  of  heat  is  maintained  at  some  place  in  the 
body,  the  steady  state  which  is  gradually  approached  may  be  one  in  which 
the  temperature  varies  from  point  to  point  but  remains  constant  at  each 
point. 

A  hot  body  is  not  like  a  pendulum  swinging  in  air  and  performing  a 
series  of  damped  oscillations  as  the  position  of  equilibrium  is  approached, 
it  is  more  like  a  pendulum  moving  in  a  very  viscous  fluid  and  approaching 
its  position  of  equilibrium  from  one  side  only.  The  steady  state  appears, 
in  fact,  to  be  approached  without  oscillation. 

These  remarks  apply,  of  course,  to  the  phenomenon  of  conduction  of 
heat  when  there  is  no  relative  motion  (on  a  large  scale)  of  different  parts 
of  the  body.  When  a  liquid  is  heated,  a  state  of  uniform  temperature  is 
produced  largely  by  convection  currents  in  which  part  of  the  fluid  Amoves 
from  one  place  to  another  and  carries  heat  with  it.  There  are  convection 
currents  also  in  the  atmosphere  and  these  are  responsible  not  only  for  the 
diffusion  of  heat  and  water  vapour  but  also  for  a  transportation  of  momentum 
which  is  responsible  for  the  diurnal  variation  of  wind  velocity  and  other 
phenomena. 


Conduction  of  Heat  119 

A  third  process  by  which  heat  may  be  lost  or  gained  by  a  body  is  by 
the  emission  or  absorption  of  radiation.  This  process  will  be  treated  here 
as  a  surface  phenomenon  so  that  the  laws  of  emission  and  absorption  are 
expressed  as  boundary  conditions  ;  the  propagation  of  the  radiation  in  the 
intervening  space  between  two  bodies  or  between  different  parts  of  the 
same  body  is  considered  in  electromagnetic  theory.  The  mechanism  of  the 
emission  or  absorption  is  not  fully  understood  and  is  best  described  by 
means  of  the  quantum  theory  and  the  theory  of  the  electron.  The  use  of  a 
simple  boundary  condition  avoids  all  the  difficulty  and  is  sufficiently 
accurate  for  most  mathematical  investigations.  In  many  problems,  how- 
ever, radiation  need  not  be  taken  into  consideration  at  all. 

The  fundamental  hypothesis  on  which  the  mathematical  theory  of  the 
conduction  of  heat  is  based  is  that  the  rate  of  transfer  of  heat  across  a 
small  element  dS  of  a  surface  of  constant  temperature  (i.e.  an  isothermal 
surface)  is  represented  by  ™ 


where  K  is  the  thermal  conductivity  of  the  substance,  0  is  the  temperature 
in  the  neighbourhood  of  dS,  and  ~~  denotes  a  differentiation  along  the 

outward-drawn  normal  to  dS.  The  negative  sign  in  this  expression  simply 
expresses  the  fact  that  the  flow  of  heat  is  from  places  of  higher  to  places 
of  lower  temperature.  The  rate  of  transfer  of  heat  across  any  surface 
element  dv  in  time  dt  may  be  denoted  by  fvdadt,  where  the  quantity  fv  is 
called  the  flux  of  heat  across  the  element  and  the  suffix  v  is  used  to  indicate 
the  direction  of  the  normal  to  the  element. 

Let  us  now  consider  a  small  tetrahedron  DABC  whose  faces  DBC, 
DC  A,  DAB,  ABC  are  normal  respectively  to  the  directions  Ox,  Oy,  Oz,  Ow, 
where  the  first  three  lines  are  parallel  to  the  axes  of  co-ordinates.  Denoting 
the  area  ABC  by  A,  the  areas  DBC,  DC  A,  DAB  are  respectively  wx&, 
Wy&,  wzA,  where  wx,  wy,  wz  are  the  direction  cosines  of  Ow. 

Wher  A  is  very  small  the  rate  at  which  heat  is  being  gained  by  the 
tetrahedron  at  time  t  is  approximately 

(wxfx  +  wvfy  +  wjz  -  /„)  A. 

7/J 

This  must  be  equal  to.  Vcp  ^~  ,  where  V  is  the  volume  of  the  tetrahedron, 

u/t 

c  the  specific  heat  of  the  material  and  p  its  density.  Now  V  =  £jpA,  where 
p  is  the  perpendicular  distance  of  D  from  the  plane  ABC,  hence 

Wxfx  +  Wyfv  +  Wzfz  -  /«  =  $PCP  ^ 

and  so  tends  to  zero  as  p  tends  to  zero. 

When  DAB  is  an  element  of  an  isothermal  surface  we  may  use  the 


120     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 
additional  hypothesis  that  fx  and  fy  are  both  zero  and  the  equation 

giV6S  f  f 

L,  -=  wz  j  z  = 

Jw  zjz 

The  law  (A)  thus  holds  not  simply  for  an  isothermal  surface  but  for 
any  surface  separating  two  portions  of  the  same  material.  The  vector  A0 

whose  components  are  x-  -  ,  ~    ,  ~    is  called  the  temperature  gradient  at  the 

point  (x,  y,  z)  at  time  t. 

Let  us  now  consider  a  portion  of  the  body  bounded  by  a  closed  surface 
R.  Assuming  that  fff  ,  fy  ,  fz  and  their  partial  derivatives  with  respect  to 
x,  y  and  z  are  continuous  functions  of  x,  y  and  z  for  all  points  of  the  region 
bounded  by  /V,  the  rate  at  which  this  region  is  gaining  heat  on  account  of 
the  fluxes  across  its  surface  elements  is 


Transforming  this  into  a  volume  integral  and  equating  the  result  to 

J7/3 
cp    '    dxdydz, 
(IT 

we  have  the  equation 

!  \CP  n i  —  -5~  (  &  *    '  "  -r     K  ^~  > '  —  ~>-  (  ^  ^    }  \  dxdydz. 
]jj[rdt      3x{     3x        3y       3y       3z\     3z  J         y 

This  must  hold  for  any  portion  of  the  material  that  is  bounded  by  a  simple 
closed  surface  and  this  condition  is  satisfied  if  at  each  point 

cp  (T  ._  div  (KV0)  =  0. 
a  I 

If  the  body  is  at  rest  we  can  write  ^  in  place  of  -j- ,  but  if  it  is  a  moving 

ut  (Jit/ 

Jf\ 

fluid  the  appropriate  expression  for  -7-  is 

d9     30         30        30         30 

~n  ^  bi   'I    ^  ^       f-  V  >r   -h  W  x-  , 

dt       dt          ox         cy         3z 
where  u,  v  and  w  are  the  component  velocities  of  the  medium. 

In  most  mathematical  investigations  the  medium  is  stationary  and  the 
quantities  c,  K  and  p  are  constant  in  both  space  and  time  and  the  equation 
takes  the  simple  form  ^ 


dt 

in  which  K  is  a  constant  called  the  diffusivity*.    If  at  the  point  (x,  y,  z) 
there  is  a  source  of  heat  supplying  in  time  dt  a  quantity  F  (x,  y,  z,t)  dxdydzdt 

*  This  name  was  suggested  by  Lord  Kelvin.    A  useful  table  of  the  quantities  K,  c,  p  and  x  is 
given  in  Ingersoll  and  Zobel's  Mathematical  Theory  of  Heat  Conduction  (Ginn  &  Co.,  1913). 


The  Drying  of  Wood  121 

of  heat  to  the  volume  element  dxdydz,  a  term  F  (x,  ?/,  z,  t)  must  be  added 
to  the  right-hand  side  of  the  equation. 

A  similar  equation  occurs  in  the  theory  of  diffusion ;  it  is  only  necessary 
to  replace  temperature  by  concentration  of  the  diffusing  substance  in  order 
to  obtain  the  derivation  of  the  equation  of  diffusion.  The  quantity  of 
diffusing  substance  conducted  from  place  to  place  now  corresponds  to  the 
amount  of  heat  that  is  being  conducted.  The  theory  of  diffusion  of  heat  was 
developed  by  Fourier,  that  of  a  substance  by  Fick.  In  recent  times  a 
theory  of  non-Fickian  diffusion  has  been  developed  in  which  the  coefficient 
K  is  not  a  constant.  Reference  may  be  made  to  the  work  of  L.  F.  Richard- 
son*. 

§  2-14.  An  equation  similar  to  the  equation  of  the  conduction  of  heat 
has  been  used  recently  by  Tuttle  f  in  a  theory  of  the  drying  of  wood.  It 
is  known  that  when  different  parts  of  a  piece  of  wood  are  at  different 
moisture  contents,  moisture  transfuses  from  the  wetter  to  the  drier  regions ; 
Tuttle  therefore  adopts  the  fundamental  hypothesis  that  the  rate  at  which 
transfusion  takes  place  transversely  with  respect  to  the  wood  fibres  or 
elements  is  proportional  to  the  slope  of  the  moisture  gradient. 

This  assumption  leads  to  the  equation 

d_e          dW 
di         W 

where  9  is  moisture  content  expressed  as  a  percentage  of  the  oven-dry 
weight  of  the  wood  and  It2  is  a  constant  for  the  particular  wood  and  may 
be  called  the  transfusivity  (across  the  grain)  of  the  species  of  wood  under 
consideration. 

From  actual  data  on  the  distribution  of  moisture  in  the  heartwood  of 
a  piece  of  Sitka  spruce  after  five  hours'  drying  at  a  temperature  of  160°  F. 
and  in  air  with  a  relative  humidity  of  30  %,  Tuttle  finds  by  a  computation 
that  h2  is  about  0-0053,  where  lengths  are  measured  in  inches,  time  in 
hours  and  moisture  content  in  percentage  of  dry  weight  of  wood. 

The  actual  boundary  conditions  considered  in  the  computation  were 

6  =  0  at  x  -  0,     9  =  0  at  x  - 1,     0  =  00  when  t  -  0. 

A  more  complete  theory  of  drying  has  been  given  recently  by  E.  E. 
LibmanJ  in  his  theory  of  porous  flow.  He  denotes  the  mass  of  fluid  per 
unit  mass  of  dry  material  by  v  and  calls  it  the  moisture  density.  The 
symbols  p,  a,  r  are  used  to  denote  the  densities  of  moist  material,  dry 
material  and  fluid  respectively  and  ft  is  used  to  denote  the  coefficient  of 
compressibility  of  the  moist  material. 

*  Proc.  Roy.  Soc.  London,  vol.  ex,  p.  709  (1926). 

f  F.  Tuttle,  Journ.  of  the  Franklin  Inst.  vol.  cc,  p.  609  ( 1925). 

t  E.  E.  Libman,  Phil.  Mttg.  (7),  vol.  iv,  p.  1285  (1927). 


122    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

The  rate  of  gain  of  fluid  per  unit  mass  of  dry  material  in  the  volume  V 
is  the  rate  of  increase  of  v,  where  v  is  the  average  value  of  v  in  F.  If  w 
is  the  mass  of  dry  material  in  volume  V  of  moist  material  and  fn  =  mass 
of  fluid  flowing  in  unit  time  across  unit  area  normal  to  the  direction  n  we 

have 


dv          V 
therefore  ~-.  =  ---  div/.  ......  (B) 

ot          w 

• 

Now,  the  mass  of  fluid  in  the  volume  V  is  wv  and  the  total  mass  of 
material  in  V  is  wv  -j-  w  and  is  also  pF,  hence 


and  (B)  gives  the  equation  < 

'dv  I  +  v 

3t=~    >" 
for  the  interior  of  the  porous  body. 

If  EdS  denotes  the  mass  of  fluid  evaporating  in  unit  time  from  a  small 
area  dS  of  the  boundary  of  the  porous  body  the  boundary  condition  is 

fn  =  E. 

The  flow  of  fluid  in  a  porous  material  may  be  regarded  as  the  sum  of 
three  separate  flows  due  respectively  to  capillarity,  gravity  and  a  pressure 
gradient  caused  by  shrinkage.  We  therefore  write,  for  the  case  in  which 
the  z  axis  is  vertical  and  p  is  the  pressure, 

-  „  dv      j       dz       7  dp 

/.--Kfr-kgrfc-kft, 

where  K  and  k  are  constants  characteristic  of  the  material. 

\-\-v 
Consider  now  a  small  element  of  volume   —  —  8w  at  the  point  P  (#,  y,  z), 

the  associated  mass  of  dry  material  being  8w  and  the  volume  per  unit  mass 
of  dry  material  1  +  v 


Then  "-  =  " 

dv      dv 

dp      dp  dV      dp  d  /I  -j-  v\ 
fo~WdvssdVdv(~j~)' 

1  dV 

But  by  definition  B  =  —  ^7  ~T-  , 

K  ap 

therefore  $=-    l    d  f1  +  ^-      !  d  /--  X  +  v 


—  -  -j-   _ 
Vfidv\    p 

1  rf  A      1  + 


\  1  d  A  1  -f  v\ 
)  =  _  -  (w  —  IU  .  ) 
)  fidv\  6  p  /' 


The  Heating  of  a  Porous  Body  123 

™  ^-  d<f>      „ 

Putting  dl-X- 

we  have  ..- 


or  /•  -  -  f  --  f  -  -  kar 

Jx~      dx'    Jv~      dy'    h~      c!z~    J  ' 

div/=  -  V*<f>, 

and  so  tp      ^  =  W, 

1  +  t>  3<          r 

wliile  the  boundary  condition  takes  the  form 

*-  +  IS  +  tgr*  =  0. 


It  should  be  mentioned  that  in  the  derivation  of  this  equation  the 
material  has  been  assumed  to  be  isotropic. 
In  the  special  case  of  no  shrinkage  we  have 

p  =  a  (1  -f  v),      ,-  =  K  ,     cf>  =  JSTi;  4-  const., 
and  the  equation  for  v  becomes 


which  is  similar  in  form  to  the  equation  of  the  conduction  of  heat.  The 

boundary  condition  is  ^  ~ 

,r  ov       n      7      oz 
A"       +  E  4-  kgr       -  0. 
dn  on 

§  2-15.  The  heating  of  a  porous  body  by  a  warm  fluid*.  A  warm  fluid 
carrying  heat  is  supposed  to  flow  with  constant  velocity  into  a  tube  which 
contains  a  porous  substance  such  as  a  solid  body  in  a  finely  divided  state. 
For  convenience  we  shall  call  the  fluid  steam  and  the  porous  substance 
iron.  The  steam  is  initially  at  a  constant  temperature  which  is  higher  than 
that  of  the  iron.  The  problem  is  to  determine  the  temperatures  of  the  iron 
and  steam  at  a  given  time  and  position  on  the  assumption  that  the  specific 
heats  of  the  iron  and  steam  are  both  constant  and  that  there  are  no*  heat 
exchanges  between  the  wall  of  the  tube  and  either  the  iron  or  steam,  no 
heat  exchanges  between  different  particles  of  steam  and  no  heat  exchanges 
between  different  particles  of  iron.  The  problem  is,  of  course,  idealised  by 
these  simplifying  assumptions.  We  make  the  further  assumption  that  the 
velocity  of  the  steam  is  the  same  all  over  the  cross-section  of  the  pipe. 
This,  too,  would  not  be  quite  true  in  actual  practice. 

Let  U  be  the  temperature  of  the  iron  at  a  place  specified  by  a  co-ordinate 
x  measured  parallel  to  the  axis  of  the  pipe,  V  the  corresponding  temperature 

*  A.  Anzelius,  Zeit*.  f.  ang.  Math.  u.  Mech.  Bd.  vi,  S.  291  (1926). 


124     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

of  the  steam.  These  quantities  will  be  regarded  as  functions  of  x  and  I  only. 
This  is  approximately  true  if  the  pipe  is  of  uniform  section  so  that  the 
cross-sectional  area  is  a  constant  quantity  A. 

Let  us  now  consider  a  slice  of  the  pipe  bounded  by  the  wall  and  two 
transverse  planes  x  and  x  f-  dx.  At  times  t  and  t  -f  dt  the  heat  contents 
of  the  iron  contained  in  this  slice  are  respectively 

uUA  dx     and     u  (  U  +  ~    dt]  A  dx, 

V  at       J 

where  u  is  the  quantity  of  heat  necessary  to  raise  the  temperature  of  unit 
volume  of  the  iron  through  unit  temperature.  Thus  the  quantity  of  heat 
imparted  to  the  iron  in  the  slice  in  time  dt  is 

dQ,=  uA~dtdx. 
ct 

Similarly,  at  time  t  the  heat  content  of  the  vapour  in  the  slice  is  rVA  dx 

(          dV     \ 
and  at  time  /  h  dt  it  is  v  (  V  -f  -^  -  dt]  A  dx,  where  v  is  a  quantity  analogous 

to  u. 

With  the  steam  flowing  across  the  plane  x  in  time  dt  a  quantity  of  heat 
vVAcdt  is  brought  into  the  slice  where  c  is  the  constant  velocity  of  flow. 

In  the  same  time  a  quantity  of  heat  v[V+  -—  dx]Acdt  leaves  the  slice  across 

V  ox       / 

the  plane  x  -f  <tx.  The  steam  has  thus  conveyed  to  the  iron  a  quantity  of 

heat  ~ir         -  Tr 

i^  (3V        5V\   A  7   7 

dQ2  =  --  v     -,-  -I-  c  ~    }A  dtdx. 

\  ot  ox/ 

In  accordance  with  the  law  of  heat  transfer  that  is  usually  adopted  the 
quantity  of  heat  transferred  from  the  steam  to  the  iron  in  the  slice  in  time 

dils  d#3  =  k(V-  U)  A  dtdx, 

where  k  is  the  heat  transfer  factor  for  iron  and  steam.   We  thus  have  the 

equations 


With  the  notation  a  =  k/cv,  b  =  k/cu  and  the  new  variables 

£  =  ax,     r  =  b  (ct  —  x), 
the  equations  become 

W-u-v    du-v-u 

3{~  '      ST       y 

These  equations  imply  that  the  quantity  A  (£,  r)  defined  by 

b(S,T)   =   #**(V-U) 

is  a  solution  of  the  partial  differential  equation 


Laplace's  Method  125 

The  supplementary  conditions  which  will  be  adopted  are 

tf  (£0)=  U19      7(0,  T)=  F1? 

where  D^  and  Fx  are  constants.  The  equation  (A)  then  gives 
f/(0,  r)=  ^-(Fi-  1^)6-', 
F(£,0)  =  C/x+CFj-  tfje-f, 

and  so  the  supplementary  conditions  for  the  quantity  A  are 
A  (f  ,  0)  =  A  (0,  r)  =  F!  -  t/!  =  IF,  say. 

§  2-16.    Solution  by  the  metfwd  of  Laplace.  The  equation  (A)  may  be 
solved  by  a  method  of  successive  approximations  by  writing 

A-  AO  +  Ax  +  A2+  ..., 

where  \  =  W  and  ~     *-  =  A^^. 

This  gives  A  -  JF/0  [2  V(fr)J, 

V  -  U=  TFe-<^>/ 


rax 

=.  Vl  -  H'c-6  <«*-*>       e~s  /0  [2  V{^  (c^  -  x)}]  ds, 

Jo 

=  Ul  +  TFe~a^  f    C<  ^c-'/o  [2 

J  o 


u  =  c/i  + 

For  x>  ct  the  solution  has  no  physical  meaning  but  for  such  values  of  oc 
the  iron  has  not  yet  been  reached  by  the  steam  and  so  U  —  C^. 

As  t  ->  oo  we  should  have  U  -+  V19  V  ->  V1 ;  this  condition  is  easily  seen 
to  be  satisfied,  for  our  formula  for  V  —  U  indicates  that  V  —  U  ->  0  and 
U  ->  Ft  because 

[    e~s  70  [2^/(axs)]  ds  =  eax. 
Jo 

The  properties  of  the  solution  might  be  used,  however,  to  infer  the  value 

of  this  integral. 

EXAMPLE 

Prove  that  if  E  (r)  =  2  -^-— 3 , 

the  differential  equation  ^—^—^    =  V 

^  .  dxdydz 

fV  (z 
is  satisfied  by      V=\     I    </>(v, w)E{x(y  —  v)(z  —  w)}dvdw 

[z    (x 
-f  I          i/i(w,u)  E  {y(z —  w)(x  —  u)}dwdu 

[x    fy 

Jo  Jo 

+  IXP  (u)  E  {(x  -  u)  yz}  du  +  (VQ(v)  E  {(y  -  v)  zx}  dv 
J  0  J  0 

+  [ZE  (w)  E  {(z  -  w)  xy}  dw  +  SE  (xyz). 
[T.  W.  Chaundy,  Proc.  London  Math.  Soc.  (2),  vol.  xxi,  p.  214  (1923).] 


126     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

§  2-21.    Riemanris  method.  Let  L  (u)  be  used  to  denote  the  differential 
expression  a 

02U  CU        ,  OU 

~   a    4-  #0    +  63    +  cuy 

dxdy         dx         dy 

where  a,  b,  c  are  continuous  (Z),  1)  in  a  region  ^?  in  the  (x,  y)  plane.   The 
adjoint  expression  L  (v)  is  defined  by  the  relation 


where  M  and  N  are  certain  quantities  which  can  be  expressed  in  terms  of 
u,  v  and  their  first  derivatives.   Appropriate  forms  for  L,  M  and  N  are 

S2v        d  d 

L  (v)  =  -     --  ^    (at;)  -  ~    (6v)  +  cv, 
dxdy     9^;  ty 

nf  I  /    du         dv\       19,     v         ^  ,  v 

If  =  aw  +v-u        =          (uv)  -  uP  (v),  say, 


'w         9i; 

2    aa.  -  « 

,  T>  7  x      9v  ^  7  .       dv  ^  , 

where        P  (v)  =  -    -  av,     Q  (v)  =  x    —60;. 

Now  if  C'  is  a  closed  curve  whiclMi^s  entirely  within  the  region  R  and 
if  both  u  and  v  are  continuous  (D,  1)  in  ,&,  we  have  by  the  two-dimensional 
form  of  Green's  theorem 


(IM  +  mN)  ds^  +         dxdy  =      [vL  (u)  -  ul  (v)]  dxdy, 

where  /,  m  are  the  direction  cosines  of  the  normal  to  the  curve  C  and  the 
double  integral  is  taken  over  the  area  bounded  by  C,  and  so  will  be  ex- 
pressed in  terms  of  the  values  of  u  and  its  normal  derivative  at  points  of 
the  curve  F,  for  when  u  is  known  its  tangential  derivative  is  known  and 

^  -  and  XT-  can  be  expressed  in  terms  of  the  normal  and  tangential  de- 

rivatives. If  (XQ,  y0)  are  the  co-ordinates  of  the  point  A  the  function  v 
which  enables  us  to  solve  the  foregoing  problem  may  be  written  in  the 

form  ,  x 

v  =  g  (x,  ?/;  x0,  #0), 

and  may  be  called  a  Green's  function  of  the  differential  expression  L  (u). 

This  theorem  will  now  be  applied  in  the  case  where  the  curve  C  consists 
of  lines  XA,  A  Y  parallel  to  the  axes  of  x  and  y  respectively  and  a  curve  I1 
joining  the  points  Y  and  X. 

Using  letters  instead  of  particular  values  of  the  variable  of  integration 
to  denote  the  end  points  of  each  integral,  we  have  when  L  (u)  =  0,  L  (v)  =  0, 


tXNdx-  \ 

J  A  J 


Riemann's  Method  127 

[A  [A 

Now          —       M  dy  =  4  \(uv)Y  —  (uv)A]  +       uP  (v)  dy, 

JY  JY 

f A  f x 

and  Ndx  =  J  [(w)x  -  (uv)  4]  -  uQ  (v)  dx, 

JA  J  A 

rx 
therefore  (uv)A  =  i  [(w)v  +  (wv)r]  +  (IM  +.mN)  efe 

J  Y 

CA  rx 

-f       ?/P  (v)  dy  —       wQ  (v)  eu-. 
J  i'  .'  .4 

If  now  the  function  v  can  be  chosen  so  that  P  (v)  =  0  on  ^4  7  and  Q  (v)  =  0 
on  AX,  the  value  of  ^  at  the  point  A  will  be  given  by  the  formula 

rx 

(uv)A  =  J  .[(uv)  v  +  (^')r]  -f        (J-W  +  mN)  ds\ 

i  Y 

It  should  be  noticed  that  if  u  is  not  a  solution  of  L  (u)  ~  0  but  a  solution  of 


the  corresponding  expression  for  u  is 

(uv)A  =  $  [(uv)x  +  (uv)Y]  +  I    (IM  -h  mN)  ds  -f-  |  U/(.r,  ? 

r 

An  interesting  property  of  the  function  y  may  be  obtained  by  consider- 
ing the  case  when  the  curve  F  consists  of  a  line  YB  parallel  to  AX  and  a 
line  BX  parallel  to  YA.  We  then  have  x 

[A  (IM  -f  mN)  ds  =  \XMdy-  {"  Ndx, 

J  Y  J  B  J  Y 


also  M  =  -  ^  V  (wv)  +  ^^  M,     ^  (w)  =  a^  + 

N^~l  L(uv]  +  "Q  (u]-  Q  (u]  =  ^x+ 

[B  [B   - 

we  have  Ndx  =  \  [(uv)Y  —  (uv)E]  +       vQ  (u)  dx, 

J  Y  J  Y 


J 


Mdy  -  |  [(uv)B  -  (uv)x]  +  f-  vP  (u)  dy. 
B  J  B 


CB  „  r 

Hence  (^^)x  =  (uv)B  —       vQ  (u)  dx  + 

JY  J 


Now  let  a  function  u  =  h  (x,y\xl,  y^)  be  supposed  to  exist  such  that 
Z/  (w)  =  0,  P  (w)  =  0  on  JBX,  Q  (u)  =  0  on  J?7  ;  the  co-ordinates  a?!,  ^  being 
those  of  B.  The  formula  then  gives 

(uv)A  -  (ttt;)^. 

Choosing  the  arbitrary  constant  multipliers  which  occur  in  the  general 
expressions  for  g  and  h,  in  such  a  way  that 

9  (z0>  y0l  x0t  y0)  -  1,     h  (xlf  y^  xl9  yj  =  1, 


128    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 
the  preceding  relation  can  be  written  in  the  form 

h  (x0,  y0;  xl9  yj  =  g  (xly  y^  XQ,  yQ). 

When  considered  as  a  function  of  (x,  y)  the  Green's  function  g  satisfies 
the  adjoint  equation  L  (v)  =  0,  but  when  considered  as  a  function  of 
(XQ,  yQ)  it  satisfies  the  original  equation  expressed  in  the  variables  xQy  yQ. 

EXAMPLE 

Prove  that  the  Green's  function  for  the  differential  equation 


is 

and  obtain  Laplace's  formula 


tt=  f*/ 

Jo 


for  a  solution  which  satisfies  the  conditions 


.5-  =  </>  (x)     when  x  =  0. 
oy 

§  2-22.    Solution  of  the  equation  ^ 


Let  the  curve  B  consist  of  a  line  A1A2  parallel  to  the  axis  of  x  and  two 
curves  Cl7  C2  starting  from  Aly  Az  respectively  and  running  in  an  upward 
direction  from  the  line  A1A2.  Let  S  denote  the  realm  bounded  by  the 
portion  of  B  below  a  line  y  parallel  to  the  axis  of  x  and  the  portion  of  B 
which  lies  between  Cl  and  (72.  When  y  is  replaced  by  the  parallel  lines 
y0,  y  the  corresponding  realms  will  be  denoted  by  SQ  and  8'  respectively. 
The  portions  of  B  below  the  lines  y,  y0  ,  y  will  be  denoted  by  /?,  /?0  and  ft' 
respectively.  The  equation  of  A1A2  will  be  taken  to  be  y  =  ylt 

We  shall  now  suppose  that  y  and  y'  both  lie  below  y0  and  that  z  is  a 

solution  of  (A)  which  is  regular  in  S0,  regularity  meaning  that  z,  x-*,  ~  - 

OX     01/ 

a2- 

and  ,5-j  are  continuous  functions  of  x  and  y  in  the  realm  S0  . 

OX 

The  differential  expression  adjoint  to  L  (z)  is  L  (f),  where 


and  we  have  the  identity 

t  [L  (,)  -/(*,  „)]  -  ZL  (0  =        «      - 


Generalised  Equation  of  Conduction  129 

Hence  if.  L  (z)  =  f  (x,  y), 

and  L  (t)  =  0, 

we  have    ^  |  [«  |  -  **]  -  |  [to]  -  «/-  0. 

Let  us  now  write  £  =  2  (£,  77),.  r  =  £  (£,  T;)  and  integrate  the  last  equation 
over  the  region  S',  then 

/,  ««-  Jr  [*«  +  Mr  '*)*•]  -//, 

In  this  equation  we  write 

r  (£  ,)  =  T  [*,  y;  f,  77]  EE  (y  -  77)-*  e^p  [-  (*  -  g)*/(y  -  77)], 
and  we  take  y  to  be  a  line  which  lies  just  below  the  line  y  which  passes 
through  the  point  (x,  y).   Our  aim  now  is  to  find  the  limiting  value  of  the 
integral  on  the  left  as  y  ->  y. 

By  means  of  the  substitution  %  =  x  -+-  2u\/(y  —  77)  this  integral  is 
transformed  into 

/•tta 

2        s  [a:  -f  2tt  (y  -  17)*,  77]  e-wt^. 

.'  MI 

If  the  equations  of  the  curves  Ol  ,  <72  are  respectively 

ar  =  cx  (y),     a;  -  c2  (y), 
the  limits  of  the  integral  are  respectively 

^i  =  [Ci  (17)  -  a;]/2  V(y  ~  T?),     ^2  =  [Cz  (>?)  -  x]/2  ^/(y  -  y). 
If  the  point  (x,  y)  lies  within  S  we  have 

ut  ->  —  oo,     i^2  -*•  +  oo  as  yx  ->  y  and  17  -+  y  ; 
if  it  lies  outside  $  we  have 

u±  ->  w2  ->  ±  oo  as  yx  -v  y  and  ?}  -+  y. 

Finally,  if  the  point  (x,  y)  lies  on  either  'C1  or  (72  one  limit  is  zero,  thus 
we  may  have  either  ut  ->  0,  u2  -+  oo  ,  or  u±  ->  —  oo,  w2  -^  0. 

The  Umiting  value  obtained  by  putting  rj  ~  yin  the  integral  is  2z  (x,  y)\/7r 
in  the  first  case,  zero  in  the  second  case  and  z  (x,  y)  \Ar  in  the  third.  Hence 
when  the  limiting  value  is  actually  attained  we  have  the  formula 

z  (xy  y)  [2V",  0  or 


VI  =  f   \T£df  +  (T  §  -  £  ^)  ^1  -  f  f 

^  L  \    ^?         c/f  /      J      .-Js 


The  transition  to  the  limit  has  been  carefully  examined  by  Levi*, 
Goursatf  and  Gevreyf.    The  last  named  has  imposed  further  conditions 

*  E.  E.  Levi,  AnnaLi  di  Matematica  (3),  vol.  xiv,  p.  187  (1908). 

t  E.  Goursat,  TraiU  #  Analyse,  t.  in,  p.  310. 

t  M.  Gevrey,  Jvurn.  de  Mathdmatiques  (6),  t.  ix,  p.  305  (1913).  See  also  Wera  Lebedeff,  Diss. 
Oottingen  (1906);  E.  Holmgren,  Arkiv  for  Matematik,  Astronomi  och  Fyaik,  Bd.  in  (1907),  Bd.  iv 
(1908);  G.  C.  Evans,  Amer.  Journ.  Math.  vol.  xxxvn,  p.  431  (1915). 


130    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

in  order  to  establish  the  formula  in  the  case  when  (x,  y)  lies  on  either  Cl 
or  C2  .  His  conditions  are  that,  if  (f>  (s)  is  continuous, 

lim  [c,  (y)  -  c,  (rj)]  (y  -  ^  =  0     (p  =  1,  2), 

r)->y 

that 

f  [c,  (y)  -  c,  (*)]  (y  -  *)-*  <f>  (s)  ds  .  exp  [-  {cp  (y)  -  cp  (s)}*/4  (y  -  s)} 

-'  >) 

should  exist  and  that  the  functions  q  (y),  c2  (y)  be  of  bounded  variation. 
It  may  be  remarked  that  the  line  integrals  in  this  formula  are  particular 

solutions  of  the  equation  ~    =  ~  2,  while  the  integral 


_  I  I     ?  (x,  y; 

/  7T  .'  „' 


,  _ 

^  \/  7T  .'  „'  >o 

is  a  particular  solution  of  the  equation  (A).  Sufficient  conditions  that  this 
may  be  true  have  been  given  by  Levi  and  less  restrictive  conditions  have 
been  formulated  by  Gevrey.  The  properties  of  the  integrals 

f  f    dT 

I(x,y)  =      T(x)y^)r])(f>(7])dr))    J(x,y)  =       ~    <f>  (77)  dy 
.  o  •  c  ^"^ 

have  also  been  studied,  where  O  is  a  curve  running  from  a  point  on  the 
line  y  =  yl  to  a  point  on  the  line  y  =  y.  It  appears  that  when  the  point  P 
crosses  the  curve  C  at  a  point  P0  the  integral  /  suffers  a  discontinuity 
indicated  by  the  formula 

lim   (JP  -  JP)  -  ±  <f>Po  VTT, 
P->PO 

the  sign  being  +  or  —  according  as  P  approaches  P0  from  the  right  or  the 
left  of  the  curve  C. 

In  this  formula  </>  denotes  any  continuous  function  and  a  suffix  P  is 
used  to  denote  the  value  of  a  function  of  position  at  the  point  'P. 

A  Green's  function  for  the  region  8  may  be  defined  by  the  formula 

G  (x,  y;  €,-n)  =  T  (x,  y;  f9r))-H  (x,  y;  £  77), 

r)%Pf        T^T-f 

where  H  (x,  y;  £,  ??),  which  satisfies  the  equation  -^  +-*—  =  0,  is  zero  on 

c£       or) 
y  when  considered  as  a  function  of  f  and  77,  which  is  regular  and  which 

takes  the  same  values  as  T  (x,  y  ;  £,  T?)  on  the  curves  Ct  and  C2  .  The  function 

n     *•  «     *u    .  x-        92#     W   3*G  ,  SG     A   . 

G  satisfies  the  two  equations  ^  =  -    ,  -x>2-  +  ^—  =  0,  is  zero  when  x  =  cl(y)y 

when  ^  =  Cj  (77),  when  x  =  c2  (y)  and  when  f  =  c2  (77),  and  is  positive  in  $. 
With  the  aid  of  this  function  a  formula 


-ff 

J  J  S 


The  Green's  Function  131 

may  be  given  for  a  solution  of  (A)  which  takes  assigned  values  on  /?.  The 
problem  of  determining  O  is  reduced  by  Gevrey  to  the  solution  of  some 
integral  equations. 

Fundamental  solutions  of  the  equations 

dz  _  a3z        d*z  _  d*z 
dy      dx3'      dy*      dx* 

have  been  obtained  by  H.  Block*  and  have  been  used  by  E.  Del  Vecchiof 
to  obtain  solutions  of  the  equations 

dz      d*z_f  d*z      d*z  _f 

dy  ~dx*~J  (X'  y)'     dy*  ^8x*~J  (X>  y}' 

EXAMPLES 
1.    Show  by  means  of  the  substitutions 


that  the  integral 

'  '  (x  -  f)*(y  -  ?)-«exp[-  (x  - 


II. 


has  a  meaning  when  p  +  1  >  0  and  p  —  2q  +  3  >  0,  /  being  an  integrable  function  . 

[E.  E.  Levi.] 

2.  Show  that  by  means  of  a  transformation  of  variables 

x'  =  x'  (x,  y),     y'  =  ±y 

the  parabolic  eq  nation  ~  -f  a  ~    -f  6  -    4-  cz  -f  /  =  0 

^  n  dx2         dx         dy 

may  be  reduced  to  the  canonical  form 

a22     dz       dz 

a  r  /2  —  a  ,  =  a  ^~,  +  cz  +f. 
dx2      dy         dx 

Show  also  that  the  term  p  ,  may  be  removed  by  making  a  substitution  of  type  z  —  uv 
ox 

and  that  the  term  involving  u  will  disappear  at  the  same  time  if 

d'2a         da  da      rt  dc 
a-  /2  =  a  a  ,  5-  ,  -f  2  .  ,  . 
dx  2        dx  dy         dx 

3.  If  a,  b  and  c  are  continuous  functions  in  a  region  R  a  solution  of 

0         (6<0) 


dx2        dx        dy 

which  is  regular  in  R  can  have  neither  a  positive  maximum  nor  a  negative  minimum.  Hence 
show  that  there  is  only  one  solution  of  the  equation  which  is  regular  in  R  and  has  assigned 
values  at  points  of  a  closed  curve  C  lying  entirely  within  R. 

[M.  Gevrey.] 

§  2-23.  Green's  theorem  for  a  general  linear  differential  equation  of  the 
second  order.  Let  the  independent  variables  xlf  x2)  ...  xm  be  regarded  as 
rectangular  co-ordinates  in  a  space  of  ra  dimensions.  The  derivatives  of 

*  Arkivf.  Mat.,  Ast.  och  Fysik,  vol.  vn  (1912),  vol.  vm  (1913),  vol.  ix  (1913). 
t  Mem.  d.  R.  Accad.  d.  Sc.  di  Torino  (2),  vol.  LXVI  (1916). 

9-2 


132     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 
a  function  u  with  respect  to  the  co-ordinates  may  be  indicated  by  suffixes 

^\  ^2 

written  outside  a  bracket,  thus  (u)2  stands  for  x     and  (u)23  for  •5—^ —  . 

(7it*2  @&2  ^*^3 

We  now  consider  the  differential  equation 

L  (u)  =  S    S  ^rs  Mr,  +  S  £r  (tt)r  -f  Fw  =  0, 

>=ls-l  r-1 

/I       _    J 

•^rs         -^sr* 

where  the  coefficients  Ars,  Br,  F  are  functions  of  xl9  x2,  ...  xm. 
The  expression  Z  (v)  adjoint  to  L  (u)  is 

L(v)=   S    2  (4rsv)rf-   S(flrt;)r  +  JTt;, 

r-l s»l  r-1 

and  we  have  the  identity 

vL(u)-uL(v)=   S  (^r)r, 

r-l 

m  77t  r  m 

where     Qr  =  -  u  2  ^4rs  (v\s  +  v  S  -4rs  (u),  4-  ^v    £r—   S  (,4rs 


The  ^-dimensional  form  of  the  theorem  for  transforming  a  surface 
integral  into  a  volume  integral  may  be  written  in  the  form 

rr  m  rr    m 

\  \  V(Qr)r  dx, ...  dxm=-\\  ZnrQrdS, 

where  nl9  n2,  ...  nm  are  the  direction  cosines  of  the  normal  to  the  hyper- 
surface  S,  the  normal  being  drawn  into  the  region  of  integration.  Hence 
we  have  the  equation 

f  f  —  r  r 

[vL  (u)  -  uL  (v)]  dxl ...  dxn  =  —  \  \{v  Dnu  -  uDnv  —  uvPn]  dS9 

vi       m 

where  Dn  (u)  =    £    2  nsArs  (u)r, 

m   r  m  n 

in 

Let  us  write  S  nsArs  =  A^r, 

where  vl9  v2,  ...  vm  are  the  direction  cosines  of  a  line  which  may  be  called 
the  conormal*,  then 

Dn  (u)  —  A  2  vr  (u)r  =  A  (u)v. 

r-l 

§  2-24.  The  characteristics  of  a  partial  differential  equation  of  the  second 
order.  Let  the  values  of  the  first  derivatives  (u)l9  (u)2,  ...  (u)m  be  given  at 
points  of  the  hypersurface  6  (xl9  x2,  ...  xm)  =  0.  If  dxl9  dx2,  ...  dxm  are 
increments  connected  by  the  equation 

(9)idxl  +  (0)2  dx2  4-  ...  (0)m  rfa?w  =  0, 
*  This  is  a  term  introduced  by  R.  d'Adh^mar. 


Characteristics  133 

and  if  (0^  ^  0,  we  may  regard  the  increments  dx2 ,  dx3 ,  ...  dxm  as  arbitrary, 

and  since  ,  r/  x  n      ,  v     ,          ,   v     ,  ,   .       7 

=  (w^a-rj  -f  (u)p2dx2  -f  ...  (M)pm&rm, 


the  quantities  (0)!  (M)^  —  (6)s  (u)pl 

may  be  regarded  as  known.   Similarly  the  quantities 

(*)i  WIP  -  (»)p  (")n 
may  be  regarded  as  known  and  so  the  quantities 


may  be  regarded  as  known.  Substituting  the  values  of  (u)ps  in  the  partial 
differential  equation 

m      m  in 

L(u)=  S    2  ^4rs  (w)r»  4-  2  Br  (u)r  -f  /V  =  0,          (I) 

r-l  s-1  r-1 

we  see  that  we  have  a  linear  equation  to  determine  (u)n  in  which  the 
coefficient  of  (u)n  is 

A  =  S    S  4r,(0)r(0),.  (II) 

If  this  quantity  is  different  from  zero  the  equation  determines  (u)n 
uniquely,  but  if  the  quantity  A  is  zero  the  equation  fails  to  determine 
(u)n  and  the  derivatives  (u)  are  likewise  not  determined.  In  this  case  the 
hypersurface  6  (x11  x2,  ...  xm)  =  0  is  called  a  characteristic  and  the  dif- 
ferential equation  A  =  0  is  called  the  partial  differential  equation  of  the 
characteristics. 

The  equations  of  Cauchy's  characteristics  for  this  partial  differential 
equation  of  the  first  order  are 

dxi       __      dx2  _       dxm 

and  these  are  called  the  bicharacteristics  of  the  original  partial  differential 
equation.  All  the  bicharacteristics  passing  through  a  point  (a^0,  a:2°,  ...  xm°) 
generate  a  hypersurface  or  conoid  with  a  singular  point  at  (xf,  x2°,  ...  #m°). 
When  all  the  quantities  A^  are  constants  this  conoid  is  identical  with  the 
characteristic  cone  which  is  tangent  to  all  the  characteristic  hypersurfaces 
through  the  point  (x-f*,  x2°,  ...  xm°). 

For  the  theory  of  characteristics  of  equations  of  higher  order  reference 
may  be  made  to  papers  by  Levi*  and  Sanniaf.  These  authors  have  also 
considered  multiple  characteristics  and  Sannia  gives  a  complete  classifica- 
tion of  linear  partial  differential  equations  in  two  variables  of  orders  up  to  5. 

*  E.  E.  Levi,  Ann.  di  Mat.  (3  a),  vol.  xvi,  p.  161  (1909). 

t  G.  Sannia,  Mem.  d.  R.  Ace.  di  Torino  (2),  vol.  LXIV  (1914);  vol.  LXVI  (1916) 


134     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

§  2-25.  The  classification  of  partial  differential  equations  of  the  second 
order.  A  partial  differential  equation  with  real  coefficients  is  said  to  be  of 
elliptic  type  when  the  quadratic  form 


is  always  positive  except  when  X1  =  X2  =  ...  ==  Xn  =  0. 

The  use  of  the  words  elliptic,  hyperbolic  and  parabolic  seems  natural  in 
the  case  n  =  2,  the  term  to  be  used  depending  upon  the  nature  of  the  conic 
AnX^  +  2A12X1X2  +  A22X2*  =  1. 

For  a  non-linear  equation  F  (r,  ,9,  t,  p,  q,  z)  =  0 

r  _  asz         3*z_    _  32z     _dz 

I  """  a*2'  '•  ""  dxdy'     ~  dy*'  P  ~~  dx' 
there  is  a  similar  classification  depending  on  the  nature  of  the  quadratic 

form 

dF  dF  dF 

\  *  _J_    9  V     X  4-    Y  2  Vr 

Al     dr  +^1^29^-+  As   -dt. 

When  n  >  2  the  classification  is  not  so  simple  ;  for  instance,  when  n  —  3 
it  may  be  based  on  the  different  types  of  quadric  surface  and  it  is  known 
that  there  are  two  different  types  of  hyperboloid. 

The  word  ellipsoidal  might  be  used  in  this  case  instead  of  elliptic,  but 
it  seems  better  to  use  the  same  term  for  all  values  of  n  because  the  im- 
portant question  from  the  standpoint  of  the  theory  of  partial  differential 
equations  is  whether  the  equation  is  or  is  not  of  elliptic  type.  For  an 
equation  of  elliptic  type  the  characteristics  are  all  imaginary  and  this  fact 
has  a  marked  influence  on  the  properties  of  the  solutions  of  the  equation. 
When  n  =  2  typical  equations  of  the  three  types  are 
d2u  d*u  du  ,  du 


d*u  du       ,  du  ^  n          i    i.  x 

~»~-  +  a  ~    4-  0  *    -f  cu  —  0  (hyperbolic), 
dxdy         dx         dy  J±  ; 

d2u         du      .  du  .  .         ,    T  , 

«  -2  -f  a  -    -h  o  ~    +  cu  =  0  (parabolic). 

A  notable  difference  between  elliptic  and  hyperbolic  equations  arises 
when  a  solution  is  required  to  assume  prescribed  values  at  points  of  a 
closed  curve  and  be  regular  within  the  curve.  For  illustration  let  us  con- 
sider the  case  when  the  curve  is  the  circle  x2  -f  y2  =  1.  If  the  boundary 
condition  is  V  —  sin  2nO  when  x  —  cos  0,  y  =  sin  6,  where  n  is  a  positive 

?2K 
integer,  there  is  no  solution  of  the  equation  ~   ~    —  0  which  is  continuous 

(D,  1)  and  single-  valued  within  and  on  the  circle*,  but  there  is  a  regular 

*  When  «.—  !  there  is  a  solution  V  =  2y  (1  —  y2)^  which  satisfies  the  boundary  condition  but 

dV 

its  derivative  —  -  is  infinite  on  the  circle. 
cy 


Classification  of  Equations  135 

927    327 
solution  of  -=-  2  +  O-T  =  °>  namely,  V  =  r2n  sin  2n0.    On  the  other  hand, 

if  the  boundary  condition  is  V  =  sin  (2n  +  1)  9,  there  is  a  solution  of 

32F  * 

-----=  0  of  type  V  =  f(y)  which  satisfies  the  conditions  and  is  single- 

valued  and  continuous  in  the  circle,  but  this  solution  is  not  unique  because 


V  =  1  —  x2  —  y2  is  a  solution  of  ~—  ~-  =  0  which  is  zero  on  the  circle  and 
y  dxdy 

single  -valued  and  continuous  inside  the  circle. 

When  the  solution  of  a  problem  is  not  unique  or  when  there  is  some 
uncertainty  regarding  the  existence  of  a  solution  the  problem  may  be 
regarded  as  not  having  been  formulated  correctly.  An  important  property 
of  the  boundary  problems  of  mathematical  physics  is  that  the  correct 
formulation  of  the  problem  is  indicated  by  the  physical  requirements  in 
nearly  every  case. 

§2-26.  A  property  of  equations  of  elliptic  type.  Picard*,  Bernsteinj 
and  Lich  tens  tern  t  have  shown  that  the  solutions  of  certain  general 
differential  equations  of  elliptic  type  cannot  have  maximum  or  minimum 
values  in  the  interior  of  a  region  within  which  they  are  regular.  This 
property,  which  has  been  known  for  a  long  time  for  the  case  of  Laplace's 
equation,  has  been  proved  recently  in  the  following  elementary  way§. 

Let  L  (u)  =  T  A^  (u)^  4-  S  Bv  (u)v 

1,1  i 

be  a  partial  differential  equation  of  the  second  order  whose  coefficients 
AHV)  Bv  are  continuous  functions  of  the  co-ordinates  (xl>  x2,  ...  xn)  of  a 
point  P  of  an  n-dimensional  region  T.  For  convenience  we  shall  sometimes 
use  a  symbol  such  as  u  (P)  to  denote  a  quantity  which  depends  on  the  co- 
ordinates of  the  point  P.  We  can  then  state  the  following  theorem  : 

//  u  (P)  is  continuous  (£),  2)  and  satisfies  the  inequality  L  (u)  >  0  every- 
where in  T,  an  inequality  of  type  u  (P)  <  u  (P0)  can  only  be  satisfied  through- 
out T,  where  P0  is  a  fixed  internal  point,  when  the  inequality  reduces  to  the 
equality  u  (P)  —  u  (P0).  Similarly,  if  L  (u)  <  0  throughout  T,  the  inequality 
u  (P)  >  u  (P0)  in  T  implies  that  u  (P)  =  u  (PQ). 

The  proof  will  be  given  for  the  case  n  =  2  so  that  we  can  use  the  familiar 
terminology  of  plane  geometry,  but  the  method  is  perfectly  general. 

Let  us  suppose  that  L  (u)  >  0  in  T  and  that  u  (P0)  =  M  ,  while  u  (P)  <  M 
if  P  is  in  T. 

If-u^M  there  will  be  a  circle  C  within  T  such  that  at  some  point  P 
of  its  boundary,  say  at  Pl9  we  have  u  (P^  =  M,  whilst  in  the  interior  of 
the  circle  u  <  M  . 

*  E.  Picard,  T  mitt  tf  Analyse,  t.  ir,  2nd  ed.,  p.  29  (Paris,  1905). 

t  S.  Bernstein,  Math.  Ann.  Bd.  ux,  S.  69  (1904). 

J  L.  Lichtenstein,  Palermo  Rend.  t.  xxxm,  p.  211  (1912);  Math.  Zeitschr.  vol.  xx,  p.  205  (1924). 

§  E.  Hopf,  Berlin.  Sitzungsber.  S.  147  (1927). 


136     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

Let  K  be  a  circular  realm  of  radius  R  whose  circular  boundary  touches 
C  internally  at  P,  then  with  the  exception  of  the  point  Pl  ,  we  have  every- 
where in  K  the  inequality  u  <  M. 

Next  let  a  circle  K1  of  radius  /^  <  R  be  drawn  so  as  to  lie  entirely 
within  T.  The  boundary  of  K±  then  consists  of  an  arc  St  (the  end  points 
included)  which  belongs  to  K  and  an  arc  $0  which  does  not  belong  to  K. 
On  St  we  have  the  inequality  u  <  M  —  e,  where  e  is  a  suitable  small 
quantity,  while  on  S0  we  have  u  ^  M.  ......  (A) 

We  now  'choose  the  centre  of  K  as  origin  and  consider  the  function 

h(P)  ==  e-'1'2-  e-i;-", 
where  r2  =  a:2  -f  ?/2  and  a  >  0.    If  xt  =  x,  x2  =  y  and 

£  (M)  -  .4w^  -f  2£?^y  -f  CM,,  -f  Dux  -f  J^, 
a  simple  calculation  gives 

e^L  (h)  =  4a2  (Ax2  -f  2&oy  -f  Cy2)  -  2a  (A  +  C  -f  Dz  -f  %). 

Since  the  equation  is  of  elliptic  type  we  have  in  the  interior  and  on  the 

boundary  of  K  .   0      rt  „  ~  9  ^   7       ^ 

J  J.r2  -f  2Bxy  -f  <7?/2  >  &  >  0, 


where  k  is  a  suitable  constant.   By  choosing  a  sufficiently  large  value  of  a 

we  can  make  r  ,  7  ,       . 

L  (//)  ^>  0 

in  Kl  and  so  L  (u  4-  8A)  >  0 

if  8  >  0.   We  have,  moreover,  h  (P)  <  0  when  P  is  on  $0,  &  (Pj)  =  0. 

.....  (B) 

We  now  put  v  (P)  =  u  (P)  -f-  8  .  h  (P),  8  >  0,  where  8  is  also  chosen  so 
small  that,  in  view  of  (A),  we  have  v  <  M  on  Sz.  On  account  of  (A)  and 
(B)  we  have  further  v  <  M  on  SQ.  Hence  v<  M  on  the  whole  of  the 
boundary  of  K1  and  at  the  centre  we  have  v  =  M  .  Thus  v  should  have  a 
maximum  value  at  some  point  in  the  interior  of  Kl  .  This,  however,  may 
be  shown  to  be  incompatible  with  the  inequality  L  (v)  >  0,  for  at  a  place 
where  v  is  a  maximum  we  have  by  the  usual  rule  of  the  differential  calculus 

0 


for  arbitrary  real  values  of  A  and  ^.   Now,  by  hypothesis, 


therefore  by  the  theorem  of  Paraf  and  Fejer  (§  1-35) 


Avxx  +  2Bvxy  +  Bvyv  <  0. 


But  the  expression  on  the  left-hand  side  is  precisely  L  (v)  since  vx  ==  vv  —  0, 
and  so  we  have  L  (v)  <  0  which  is  incompatible  with  L  (v)  >  0. 

The  case  in  which  L  (u)  <  0,  u  (P)  >  u  (P0)  can  be  treated  in  a  similar 
way. 


Maxima  and  Minima  of  Solutions  137 

In  particular,  if  L  (u)  =  0  in  T,  where  u  is  not  constant,  neither  of  the 
inequalities  u  (P)  <  u  (P0),  u  (P)  >  u  (P0)  can  hold  throughout  T  when  P 
i&  an  internal  point.  This  means  that  u  (P)  cannot  have  a  maximum  or 
minimum  value  in  the  interior  of  a  region  T  within  which  it  is  regular. 

This  theorem  has  been  extended  by  Hopf  to  the  case  in  which  the 
fr notions  A,  B,  C,  D,  E  are  not  continuous  throughout  T  but  are  bounded 
functions  such  that  an  inequality  of  type 

AX2  +  2BXfM  +  C>2  >  N  (A2  +  n*)  >  0 

holds,  with  a  suitable  value  of  the  constant  N,  for  all  real  values  of  A  and 
H  and  for  all  points  P  in  T. 

The  work  of  Picard  has  also  been  generalised  by  Moutard*  and  Fejerf. 
The  latter  gives  the  theorem  the  following  form : 

Let 

n, ...  n  n 

S    aik(x1)x2)  ...xn)(u)tk+  I>br(xl9x29  ...  xn)  (u)r  +  c  (xl9  ...xn)u=  0, 
i,...i  i 

atk  (Xl>  %2>   •••   Xn)    —  aJft  (X\>  ^2?    •••   %n) 

be  a  homogeneous  linear  partial  differential  equation  of  the  second  order 
with  real  independent  variables  xl9x2,  ...  xn  and  a  real  unknown  function 
u  (xl9  x2)  ...  xn).  The  coefficients 

^ik  \%lj  *^2>   •*•  ^n/9       ®r  \%l  j  *^2  >   •••  ^n/9       ^  (^1  >  ^2  >   •••  *^n) 

are  all  real  functions  which  can  be  expanded  in  convergent  power  series  of 

yP6S  c  (xl9  *2,  ...  xn)  -  c  +  Cj^  +  ...  cnxn  +  cnx^  -f  ... , 

6r  (a?!,  #2,  ...  xn)  =  6r  -f  &rl#i  +  ...  ftrn^n  -f  brllxf  +  .... 

«fjfc  (^1,  ^   -•  *n)   =   «tA:  +   GW^l  +    ..-, 

for  \Xi\<*i,     I  ^2  I  <  ^2»  —  I  %n  |  <  ««, 

where  zl9  z2,  ...  0n  are  suitable  constants.  Then,  if 

n,  n 

2  atkytyk  >  0 
1,1 

for  all  real  values  of  yl9  y2,  ...  yny  that  is,  if  the  quadratic  form  is  non- 
negative,  and  if  c  <  0,  the  differential  equation  has  no  solution  which  is 
regular  at  the  origin  and  has  there  either  a  negative  minimum  or  a  positive 
maximum.  If,  however,  the  quadratic  form  is  not  negative,  that  is,  if 

71,  71 

2  at*iM*  <  0 

for  some  set  of  values  rjl9  ^2,  ...  rjk9  there  is  always  a  solution  regular  at 
the  origin  which,  if  c  <  0,  has  either  a  negative  minimum  or  a  positive 
maximum.  Thus  when  c<  0  the  requirement  that  the  quadratic  form 

*  Th.  Moutard,  Journ.  de  Vtfcole  Poll/technique,  t.  LXIV,  p.  55  (1894);  see  also  A.  Paraf,  Annates 
de  Toulouse,  t.  vi,  H,  p.  1  (1892). 
t  Loc.  cit. 


138    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

should  not  be  of  the  non-negative  type  is  a  necessary  and  sufficient  con- 
dition for  the  existence  of  a  negative  minimum  or  positive  maximum  at 
the  origin  for  some  regular  solution  of  the  differential  equation. 

§  2-31.    Green's  theorem  for  Laplace's  equation.    Let  us  now  write  in 
equation  (A)  of  §  2-11 


the  equation  then  takes  the  form 


dU  dV     dU  dV     dU  dV\ 


rrdv\jv      trrwrj        t  (  d          \ 

U  v~  }dS  =     UV2Vdr  -Mis-  -x--  +  ~-  3-  4-  -5-  -~- 
on/  J  J  \ox    dx      dy    dy       dz    dz  J 

dV 
where  the  symbol  ~--  is  used  for  the  normal  component  of  VF, 

dV  dV         dV        dV 


Interchanging  U  and  V  we  have  likewise 

((vdu\jv      (VWTJ   _L  [fiUdVdUdV     dUdV 

I  K  5—  )  a£  =      y\2Udr  +  -  -=--  +    -  -  -^-  -  -  -^-  —  - 

J  \     onj  )  )  \ox   dx       dy   dy       dz    dz 

Subtracting  we  obtain  Green's  theorem, 


In  this  equation  the  functions  U  and  V  are  supposed  to  be  continuous 
(D,  2)  within  the  region  over  which  the  volume  integrals  are  taken.  This 
supposition  is  really  too  restrictive  but  it  will  be  replaced  later  by  one 
which  is  not  quite  so  restrictive.  If  the  functions  U  and  V  are  solutions 
of  the  same  differential  equation  and  one  which  has  the  same  characteristic 
as  Laplace's  equation,  an  interesting  result  is  obtained.  In  particular,  if 

k*U  -  0, 


where  k  is  either  a  constant  or  a  function  of  x,  y  and  z,  the  volume  integral 
vanishes  and  we  have  the  relation 


In  the  special  case  when  the  surface  S  is  a  sphere  and 

U  =  fm(r)Tm(**g\     V  =  fn(r)Yn(HZ}, 


where  Ym  and  Yn  are  functions  which  are  continuous  over  the  sphere  and 
fm  (r),  fn  (r)  are  f  unctions  such  that  fm  (r)  fn'  (r)  ^  fm'  (r)  fn  (r)  on  the  sphere 
we  obtain  the  important  integral  relation 


which  implies  that  the  functions  Ym  form  an  orthogonal  system. 


Green's  Theorem  for  Laplace's  Equation  139 

An  appropriate  set  of  such  functions  will  be  constructed  in  §  6-34. 
When  k  is  a  constant  the  equations  (B)  may  be  derived  from  the  wave- 
equations  D2^  =  0,  D2^  =  0  by  supposing  that  u  and  v  have  the  forms 

u  =  U  sin  (kct  -ha),     v  =  V  sin  (kct  -f  /?) 

respectively.    When  k  is  real  these  wave  -functions  are  periodic.    When 
k  =  0,  U  and  V  are  solutions  of  Laplace's  equation. 

The  equation  may  also  be  derived  from  the  equation  of  the  conduction 
of  heat, 


by  supposing  that  this  possesses  a  solution  of  type  v  =  e~*m  V(x,  y,  z). 

Green's  theorem  is  particularly  useful  for  proofs  of  the  uniqueness  of 
the  solution  of  a  boundary  problem  for  one  of  our  differential  equations. 
Suppose,  for  instance,  that  we  wish  to  find  a  solution  of  Laplace's  equation 
which  is  continuous  (Z),  2)  within  the  region  bounded  by  the  surface  S  and 
which  takes  an  assigned  value  F  (x,  y,  z)  on  the  boundary  of  S.  If  there  are 
two  such  solutions  U  and  F  the  difference  W  =  U  —  V  will  be  a  solution 
of  Laplace's  equation  which  is  zero  on  the  boundary  and  continuous  (D,  2) 
within  the  region  bounded  by  8.  Green's  theorem  now  gives 

n     fw  zw  ,a     (T/swY    /3WV    fiw\*\  i 
0  ==  \w  ---  dS  =     (  -3  -  -  )  +  K-  )  +  hr  )     dr> 
J        dn  }[\dx  J       \dyj       \  dz  )  J 

and  this  equation  implies  that 


dW_ 
y"        '      dz  ~Uj 

W  is  consequently  constant  and  therefore  equal  to  its  boundary  value  zero. 
Hence  U  =  V  an|J  the  solution  of  the  problem  is  unique.    A  similar  con- 

dW  dW 

elusion  may  be  drawn  if  the  boundary  condition  is  ^~  =0  or  -~  —  h  h  W  =  0, 

where  h  is  positive.   If  the  equation  is  V2F  -h  AF  =  0  instead  of  Laplace's 
equation  the  foregoing  argument  still  holds  when  A  is  negative,  for  we  have 

the  additional  term  —  A  I  W2dr  on  the  right-hand  side.   The  argument 

breaks  down,  however,  when  A  is  positive. 

In  the  case  of  the  equation  of  heat  conduction  (C)  there  are  some 

similar  theorems  relating  to  the  uniqueness  of  solutions.    If  possible,  let 

9?^        _ 
there  be  two  independent  solutions  vly  v2  of  the  equation  ^   =  *:V2t>  and 

the  supplementary  conditions 

v=f(x,  y,  z)  for  t  =  0  for  points  within  S, 

v  ~  <f>  (x,  y,  z,  t)  on  S  (t  >  0), 

v  continuous  (J5,  2)  within  region  bounded  by  S. 


140     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 
Let  V  =  v1  -  v2,  then  F  =  0  f or  t  =  0  within  S,  V  =  0  on  S. 

Putting  27  -  [  F2dr, 

we  have  ^  =  f  F  3/dr  =  /c  I  V  (V2F)  dr 

c#      J       9£  ' 


-I 


T/^F  70         [[fiV\*     /3F\2     /3          7 

F  .-  AS  -  *      U    )  +  br  )  +    ^      \dr' 
dn  ]  [\Sx/       \dyj       \  dz 


Since  F  =  0  on  S  the  first  integral  vanishes,  and  so 

dl  /T/^V  /3F\2  /SV\2]      7  /,^AX 

~.  -  -  AC         ----  )    -I   U,  -  )    +     ~  rfr       *  >  0). 

c#  J  [_\  do;  /       \G7//        \  c/2  /  J 

But  7  =  0  for  t  =  0,  therefore  7  <  0,  but  on  the  other  hand  the  integral 
for  7  indicates  that  7  >  0,  consequently  we  must  have  7=0,  F  =  0. 

These  theorems  prove  the  uniqueness  of  solutions  of  certain  boundary 
problems  but  they  do  not  show  that  such  solutions  exist.  Many  existence 
theorems  have  been  established  by  the  methods  of  advanced  analysis  and 
the  literature  on  this  subject  is  now  very  extensive. 

§  2-32.  Green's  functions.  The  solution  of  a  problem  in  which  a  solution 
of  Laplace's  equation  or  a  periodic  wave-function  is  to  be  determined  from 
a  knowledge  of  its  behaviour  at  certain  boundaries  can  be  made  to  depend 
on  that  of  another  problem — the  determination  of  the  appropriate  Green's 
function*. 

Let  Q(x,  y,  z-9xl9  yl,z1)  be  a  solution  of  V2G  +  k2G  =  0  with  the 
following  properties:  It  is  finite  and  continuous  (Dy  2)  with  respect  to 
either  x,  y,  z  or  xl ,  yl ,  z1  in  a  region  bounded  by  a  surface  S,  except  in  the 
neighbourhood  of  the  point  (x19  yl9  zj  where  it  is  infinite  like  T?"1  cos  kR 
as  7?  ->  0,  R  being  the  distance  between  the  points  (x,  y,&)  and  (xl9  yl^  z^). 
At  the  surface  S9  some  boundary  condition  such  as  (1)  G  —  0,  (2)  dG/dn  =  0, 
or  (3)  dG/dn  -f-  hG  =  0  is  satisfied,  h  being  a  positive  constant. 

Adopting  the  notation  of  Plemeljt  and  KneserJ,  we  shall  denote  the 
values  of  a  function  <j>  (f,  77,  £)  at  the  points  (x,  y,  z)9  (xl9  yly  zt)  respectively 
by  the  symbols  </>  (0)  and  <£(!). 

When  a  function  like  the  Green's  function  depends  upon  the  co- 
ordinates of  both  points  it  will  be  denoted  by  a  symbol  such  as  G  (0,  1). 
The  importance  of  the  Green's  function  depends  chiefly  upon  the  following 
theorem : 

Let  U  be  a  solution  of 

V2C7  +  k*U  +  47T/  (x,  y,  z)  -  0,  (A) 

*  G.  Green,  Math.  Papers,  p.  31. 

t  Monatshefte  far  Math.  u.  Phys.  Bd.  xv,  S.  337  (1904). 

J  A.  Kneser,  Die  Integralgleichungen  und  ihre  Anwendungen  in  der  mathematischen  Physik 
(Vieweg,  Brunswick,  1911). 


Green's  Functions  141 

which  is  finite  and  continuous  (Z>,  2)  throughout  the  interior  of  a  region 
®  bounded  by  a  surface  8  and  let  /  (x,  y,  z)  be  a  function  which  is  finite 
and  continuous  throughout  !3).  We  shall  also  allow  /  (x,  y,  z)  to  be  finite 
and  continuous  throughout  parts  of  the  region  and  zero  elsewhere. 

Applying  Green's  theorem  to  the  region  between  a  small  sphere  whose 
centre  is  at  (x± ,  yl ,  zj  and  the  surface  S,  we  have 


Now  V2(?  —  —  k2G  and  the  first  integral  on  the  right  may  be  found  by 
a  simple  extension  of  the  analysis  already  used  in  a  similar  case  when 
G  =  1/7?,  consequently  we  have  the  equation 

(1)  =  (O,  I)/  (0)dT0  +  --GddS0  .......  (B) 


If  U  satisfies  the  same  boundary  conditions  as  G  on  the  surface  8  the 
surface  integral  vanishes  and  we  have* 


'0.  (C) 

If,  on  the  other  hand,  /  (x,  y,  z)  =  0  and  G  =  0  on  8  we  have 

(D) 

the  value  of  U  is  thus  determined  completely  and  uniquely  by  its  boundary 

o/nr 

values.    Similarly,  if  the  boundary  condition  is  ~     =  0  on  S  we  have 

on 

?„,  (E) 


and  the  value  of  U  is  determined  by  the  boundary  values  of  dU/dn. 

Finally,  if  the  boundary  condition  satisfied  by  0  is  «-    +  hG  =  0,  we 

have 

(F) 


and  U  is  expressed  in  terms  of  the  boundary  values  of  ^-  —  f-  hU. 

If  g  (x2,  2/2,  z2;  x,  y,  z)  is  the  Green's  function  for  the  same  boundary 
condition  as  G  (0,  1)  but  for  the  value  a  of  &,  we  must  also  surround  the 

*  It  has  not  been  proved  that  whenever  the  function  /  is  finite  and  continuous  throughout  D 
the  formula  (C)  gives  a  solution  of  (A).  Petrmi  has  shown  in  fact  that  when/  is  merely  continuous 
the  second  derivatives  of  the  integral  may  not  exist  or  may  not  be  finite.  Ada  Math.  t.  xxxr,  p.  127 
(1908).  It  should  be  remarked  that  Gauss  in  1840  derived  Poisson's  equation  (§2-61)  on  the 
supposition  that  the  density  function/  is  continuous  (/>,  1).  With  this  supposition  (C)  does  give 
a  solution  of  (A).  Poisson's  equation  and  the  solution  of  (A)  are  usually  derived  now  for  the  case  of  a 
function  /  which  satisfies  a  Holder  condition.  See  Kcllogg's  Foundations  of  Potential  Theory, 
ch.  vi. 


142     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

point  (#2,  y*-,  z*)  by  a  small  sphere  when  we  apply  Green's  theorem  with 
U  (x,  y,  z)  =  g  (2,  0).   We  then  obtain  the  equation 

/£2  _  (jl^    ,' 

g  (2,  1)  =  G  (2,  1)  -  L_J    g  (2,  0)  G  (0,  1)  dr,  .......  (G) 


This  may  be  regarded  as  an  integral  equation  for  the  determination  of 
g  (2,  0)  when  G  (0,  1)  is  known  or  for  the  determination  of  G  (0,  1)  when 
g  (2,  1)  is  known.  In  some  cases  the  Green's  function  for  Laplace's  equation 
(k  =  0)  can  be  found  and  then  the  integral  equation  can  be  used  to  calculate 
g  (2,  0)  or  to  establish  its  existence.  The  Green's  function  for  Laplace's 
equation,  when  it  exists,  is  unique,  for  if  G  (0,  1),  H  (0,  1)  were  two  different 
Green's  functions  the  function 

V  (0)  =  G  (0,  1)  -  H  (0,  1) 

would  be  continuous  (D,  2)  throughout  the  region  bounded  by  the  surface 
S  and  satisfy  the  boundary  condition  that  was  assigned,  but  such  a  function 
is  known  to  be  zero. 

For  small  values  of  a2  the  function  g  (2,  0)  can  be  obtained  by  expanding 
it  in  the  form  g  (^  Q)  =  fl  (2>  Q)  +  ^  (^  Q)  +  ..........  (R) 

The  first  term  is  the  corresponding  Green's  function  for  Laplace's 
equation  and  is  known,  the  other  terms  may  be  obtained  successively  by 
substituting  the  series  in  the  integral  equation  (with  k  =  0)  and  equating 
coefficients  of  the  different  powers  of  a2.  80  long  as  the  series  converges 
this  method  gives  a  unique  value  of  g  (2,  0).  The  value  of  g  (2,  0),  if  it 
exists,  will  certainly  not  be  unique  when  a2  has  a  singular  or  characteristic 
value  for  which  the  "homogeneous  integral  equation"  has  a  solution  </> 
which  is  different  from  zero.  In  this  case 

4rr</>  (1)  =  (a2  -  i«)  JV  (0)  G  (0,  1)  rfr0,  ......  (I) 

and  the  formula  (C)  indicates  that  this  function  <£  (0)  =  U  (x,  y,  z)  is  a 
solution  of  V2£/-fcr2t/=0,  which  satisfies  the  assigned  boundary  con- 
ditions and  the  other  conditions  imposed  on  U.  The  solutions  of  this  type 
are  of  great  importance  in  many  branches  of  mathematical  physics, 
particularly  in  the  theory  of  vibrations,  and  have  been  discussed  by  many 
writers. 

The  characteristic  values  of  a2  are  called  Eigenwerte  by  the  Germans 
and  the  corresponding  functions  </>  Eigenfunktionen.  These  terms  are 
now  being  used  by  American  writers,  but  it  seems  worth  while  to  shorten 
them  and  use  eit  in  place  of  Eigenwert  and  eif  in  place  of  Eigen- 
funktion.  The  same  terms  may  be  used  also  in  connection  with  the 
homogeneous  integral  equation  (I).  In  discussing  this  equation  it  is 
convenient,  however,  to  put  k  =  0,  so  that  G  becomes  the  Green's  function 


Symmetry  of  a  Green's  Function  143 

for  Laplace's  equation  and  the  assigned  boundary  conditions.  Denoting 
this  function  by  the  symbol  4-rrK  (0,  1)  we  have  the  integral  equation 

(0)  K  (0,  1)  dr0 

for  the  determination  of  the  solution  of  V2<f>  +  cr2<f>  =  0  and  the  assigned 
boundary  conditions,  that  is,  for  the  determination  of  the  eifs  and  eits. 
The  function  K  (0,  1)  is  called  the  kernel  of  the  integral  equation;  it  has 
the  important  property  of  symmetry  expressed  by  the  equation 

K  (0,  1)  =  K  (1,  0). 

This  may  be  seen  as  follows. 

If  we  put  4rr/  (0)  =  (o-2  —  k2)  g  (0,  2)  in  the  formula  (B)  and  proceed 
as  before,  Green's  theorem  gives 

g  (I,  2)  =  0  (2,  1)  -  -^^ /</  (0,  2)  O  (0,  1)  dr0. 

Putting  o-  =  k  and  comparing  this  equation  with  the  previous  one  we 
obtain  the  desired  relation.  When  k  ^  0  the  relation  gives 

0(1,  2)  =  0(2,1). 
When  the  boundary  condition  is  ~-  =  0  this  result  is  equivalent  to  one 

(j'Yl 

given  by  Helmholtz  in  the  theory  of  sound.  If  \fj  (0)  is  an  eif  corresponding 
to  an  eit  v2  which  is  different  from  a2  we  have 


I  (1)  =  v2        (0)  K  (0,  1)  dr0, 
and,  if  the  order  of  integration  can  be  changed, 

K(0, 


=  „«    <£  (0)  I  (0)  dr.  =  v«        (1)  *  (1)  drj. 
Hence  the  eifs  <£  and  «/r  satisfy  the  orthogonal  relation 


This  result  may  be  used  to  prove  that  the  eits  a2  are  all  real.  If, 
indeed,  a2  were  a  complex  quantity  a  +  if!  the  corresponding  eif  <f>  (0) 
would  also  be  a  complex  quantity  x  (0)  +  i<*>  (0),  and  since  K  is  real  the 
function  0  (0)  =  x  (0)  —  *<*>  (0)  would  be  an  eif  corresponding  to  the  eit 
v2  =  a  —  i/J,  and  the  orthogonal  relation 


0  =  jt  (0)  0  (0)  <*TO  =  |{[X  (O)]2  +  [co  (0)]*}  dr. 
would  be  satisfied.  This,  however,  is  impossible  because  the  integrand  is 


144    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 
either  zero  for  all  values  of  the  variables  or  positive  for  some,  but  it  is  zero 
only  when  x  (0)  -  «,  (0)  -  0,     *  (0)  -  0. 

To  prove  that  the  eits  are  all  positive  we  make  use  of  the  equation 


*  j  U'dr  =  -  f  U  Vffrfr-  f  \(f-)2  4-  (^)2  +  (f  )']  dr. 
J  J  J  l\ox/        \dy  /        \dz  /  J 

The  Green's  function  is  usually  found  in  practice  by  finding  the  eifs 
and  eits  directly  from  the  differential  equation  and  then  writing  down  a 
suitable  expansion  for  G  in  terms  of  these  eifs.  The  question  of  convergence 
is,  however,  a  difficult  one  which  needs  careful  study.  The  method  has  been 
used  with  considerable  success  by  Heine  in  his  Kugelfunktionen,  by  Hilbert 
and  his  co-workers,  by  Sommerfeld,  Kneser  and  Macdonald, 

§  2-33.  Partial  difference  equations.  The  partial  difference  equations 
analogous  to  the  partial  differential  equations  satisfied  by  conjugate 
functions  are  7/  _  .,  ,,,  _  -, 

ux  —   "Hi       My  —   ~  vx> 

and  these  lead  to  the  equations  of  §  1-62 

uxx  -f-  uvv  =  0,       Vxz  4-  Vyy  =  0, 

which  are  analogous  to  Laplace's  equation.  These  difference  equations 
have  been  used  in  recent  years  to  find  approximate  solutions  of  Laplace's 
equation  when  certain  boundary  conditions  are  prescribed*  and  also  to 
establish  the  existence  of  a  solution  corresponding  to  prescribed  boundary 
conditions. 

Let  us  consider,  for  instance,  a  square  whose  sides  are  x  —  ±  2  h, 
y  —  -{_  2h  and  let  us  introduce  the  abbreviations 

a  =  h,     b  =  2h,     a  =  —  h,     j8  =  —  2h,     u  (x,  y)  =  (xy), 
u(b,y)=(y),     u(p,y)  =  (y);     u(x,b)=[x],     u  (x,  p)  =  [x], 
then  we  have  eight  non-homogeneous  equations  (fi-A-equations) 

-  (Oa)  -  («0)  +  4  (aa)  =  (a)  -f  [a],     (Oa)  +  (aO)  -  4  (aa)  -  (a)  -f  [a], 

-  («0)  -  (Oa)  -f-  4  (aa)  -  (a)  -f  [a]  ,     (Oa)  -f  (aO)  -  4  (aa)  -  (a)  +  [a], 
(aa)  +  (00)  -f  (aa)  -  4  (aO)  =  -  (0),     (act)  +  (00)  +  (aa)  -  4  (aO)  -  -  (0), 
(aa)  +  (00)  +  (aa)  -  4  (Oa)  =  -  [0],     (aa)  +  (00)  -f  (aa)  -  4  (Oa)  =  -  [0], 

and  one  homogeneous  equation  (A-equation) 

(Oa)  4-  (aO)  4-  (Oa)  4-  (aO)  -  4  (00). 

The  first  step  in  the  solution  is  to  eliminate  the  quantities  (aa),  (aa), 
(aa),  (aa)  which  do  not  occur  in  the  ^-equation.  This  gives  the  equations 
4  (00)  4-  (Oa)  4-  (Oa)  -  14  (aO)  4-  (a)  4-  (a)  4-  [a]  4-  [a]  4-  4  (6)  -  0, 
4  (00)  4-  (Oa)  4-  (Oa)  -  14  (aO)  4-  (a)  4-  (a)  4-  [a]  4-  [a]  4-  4  (0)  =  0, 
4  (00)  4-  (aO)  4-  (aO)  -  14  (Oa)  4-  (a)  4-  [a]  4-  (a)  4-  [a]  +  4  [0]  =  0, 
4  (00)  4-  (aO)  4-  (aO)  -  14  (Oa)  4-  (a)  4-  [a]  4-  (a)  +  [a]  +  4  [0]  =  0.  , 

*  L.  F.  Richardson,  Phil.  Trans.  A,  vol.  ccx,  p.  307  (1911);  Math.  Gazette  (July,  1925). 


Partial  Difference  Equations  145 

Adding  these  equations  we  have 

-  16  (00)  +  12  (aO)  -f  12  (Oa)  -f  12  (aO)  -f  12  (Oa) 
=  2  (a)  +  2  (a)  +  2  (a)  -f  2  (a)  -f  2  [a]  -f  2  [a]  -f  2  [a]  4-  2  [a] 
+  4(0)  +  4(0)  +  4[0]+4[6]. 

Combining  this  with  the  homogeneous  equation  we  see  that  the  quantity 
on  the  right-hand  side  of  the  last  equation  is  equal  to  -f  32  (00)  and  so 
the  quantity  (00)  is  obtained  uniquely. 

Similarly,  if  the  sides  of  the  square  are  x  =  ±  3h,  y  =  ±  3h  there  are 
16  7i-A-equations  and  9  A-equations  which  may  be  solved  by  first  eliminating 
the  quantities  which  do  not  occur  in  the  /^-equations.  We  have  then  to 
solve  9  linear  equations  in  order  to  obtain  the  remaining  quantities,  but 
these  9  equations  may  be  treated  in  exactly  the  same  way  as  the  previous 
set  of  9  linear  equations,  quantities  being  eliminated  which  do  not  occur 
in  the  central  equation.  In  this  way  a  value  is  finally  found  for  (00). 

A  similar  method  may  be  used  for  a  more  general  type  of  square  net- 
work or  lattice.  Let  the  four  points  (x  -f-  h,  y),  (x  —  h,  y),  (x,  y  ,+  /*)» 
(x,  y  —  h)  be  called  the  neighbours  of  the  point  (x,  y)  and  let  the  lattice  L 
consist  of  interior  points  P,  each  of  which  has  four  neighbours  belonging 
to  the  lattice,  and  boundary  points  Q,  each  of  which  has  at  least  one 
neighbour  belonging  to  the  lattice  and  at  least  one  neighbour  which  does 
not  belong  to  the  lattice.  A  chain  of  lattice  points  Alt  A2,  ...  An+1  is  said 
to  be  connected  when  ^4S+1  is  one  of  the  neighbours  of  As  for  each  value 
of  ^  in  the  series  1,  2,  ...  n.  A  lattice  L  is  said  to  be  connected  when  any 
two  of  its  points  belong  to  a  connected  chain  of  lattice  points,  whether  the 
two  points  are  interior  points  or  boundary  points.  The  lattice  has  a  simple 
boundary  when  any  two  boundary  points  belong  to  a  chain  for  which  no 
two  consecutive  points  are  internal  points  and  no  internal  point  P  is 
consecutive  to  two  boundary  points  having  the  same  x  or  the  same  y  as  P. 

The  solubility  of  the  set  of  linear  equations  represented  by  the  equation 

ux-x  -f  uyy  =  0  (A) 

for  such  a  lattice  may  be  inferred  from  the  fact  that  this  set  of  linear 
equations  is  associated  with  a  certain  quadratic  form 

h2  2  (ux*  f  uj)9 

where  the  summation  extends  over  all  the  lattice  points,  and  a  difference 
quotient  associated  with  a  boundary  point  is  regarded  as  zero  when  a  point 
not  belonging  to  the  lattice  would  be  needed  for  its  definition.  This  sum- 
mation can,  by  the  so-called  Green's  formula,  be  expressed  in  the  form 

-  WJ^u(ux2  +  uvy)  -  h^uR(u),  (B) 

P  Q 

where  the  boundary  expression  R  (u)  associated  with  a  boundary  point  UQ 
is  defined  by  the  equation 

hR  (UQ)  —  u±  -f  u2  -h  ...  us  —  &UQ, 


146    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

where  u^u^,  ...  UB  are  the  8  neighbouring  points  of  UQ  (s  <  3).  Since 
uxx  -f  uv?  =  0  the  quadratic  form  can  be  expressed  in  terms  of  boundary 
values.  If  there  were  two  solutions  of  the  partial  difference  equation  with 
the  same  boundary  values,  the  foregoing  identity  could  be  applied  to  their 
difference  u  —  vy  and  since  the  boundary  values  of  u  —  v  are  all  zero  the 
identity  would  give  the  relation 

2  [(*„  -  *.)•  +  (u,  -  vyy]  =  o, 

which  implies  that  ux  —  vx  =  0,  uy  —  vy  =  0  for  all  points  (x,  y)  of  the 
lattice ;  consequently,  since  u  —  v  is  zero  on  the  boundary  it  must  be  zero 
throughout  the  lattice. 

There  is  another  identity 

0  =  A2  2  (vuxx  -f  vUyj  —  uvxx  —  uVyy)  +  h  2  [vR  (u)  —  uR  (v)] 

P  Q 

which,  when  applied  to  the  case  in  which  uxx  -f  uyy  =  0  and  the  boundary^ 
value  of  v  is  zero,  gives 

2  [(ux  +  vx)*  +  (uy  +  vy)*]  =  -  2  (u  +  v)  (vxx  +  vvy)  -  h~l  S  (u)  [R  (v)  +  R  (u)] 

P  Q 

=  -  A-1  S  [vR  (u)  -  uR  (v)]  +  2  (vx*  +  vv2  +  ux*  +  uy2) 

Q 

+  A-1  2  uR  (u)  -  h-1  2  [uR  (u)  +  uR  (v)] 

Q  Q 

=  S  (v.»  +  vy2  +  ux*  +  uy*) 

>  S  (uj  +  uy*), 

the  transformations  being  made  with  the  aid  of  Green's  formula  (B). 

This  equation  shows  that  the  solution  of  uxx  -p  uyy  =  0  and  the  pre- 
scribed boundary  condition  gives  the  least  possible  value  to  the  quadratic 
form.  The  system  of  linear  equations  uxx  -f  uyy  =  0  can,  indeed,  be  ob- 
tained by  writing  down  the  conditions  that  the  quadratic  form  should  be 
a  minimum  when  the  boundary  values  of  u  are  assigned. 

With  a  change  of  notation  the  quadratic  form  may  be  written  in  the 
form  N  N  N 

2    2  cmnumun  -  2  2  anun  +  6, 

m»l n— 1  n— 1 

where  the  quadratic  form  is  never  negative.  The  corresponding  set  of  linear 

equations 
H  -f  ciaw2  -f-  ...  CINUN  -  a1? 


has  a  determinant  |  cmn  |  which  is  not  zero  and  so  can  be  solved. 

For  the  sake  of  illustration  we  consider  a  lattice  in  which  the  internal 
lattice  points  are  represented  in  the  diagram  by  the  corresponding  values 


Associated  Quadratic  Form  147 

of  the  variable  u  and  the  boundary  lattice  points  by  corresponding  values 
denoted  by  t/s. 


The  quadratic  form  is  in  this  case 
K-*g2+K-^i)2+K-^ 

to  -  *>e)2  4-  K  -  *4>)2  +  («o  -  ^io)2  +  K  -  ^)2  +  to  -  ^i)2  +  to  -  t>9)2, 

(t>3  -  ^2)2  +   K  -  ^8)2  4-   (K,  ~  ^4)2  +   (*4  ~  "5)2> 

and  it  is  easy  to  see  that  the  equations  obtained  by  differentiating  with 
respect  to  u±  ,  u2  respectively  are 

4^j  =  u0  +  u2  +  ^  -f  v2J     4^2  =  ^  +  1*3+^5  +  V3> 

and  are  of  the  required  type.  The  quadratic  form  is,  moreover,  equal  to 
the  sum  of  the  quantities 

t^-Wa-VM-tJoJ+M^ 
-  <NO  -  w2  -  u4  -  v9)  4-  w4  (4w4  -w3  -i*5  -  v1  -  vB)+u6  (4w6  -^2  -^4-^-^4), 

^o  («o  -  wo)  4-  Vi  («i  ~  ^)  +  v2  (^2  -  ^i)  4-  %  to  ~  ^2)  4-  ^4  (^4  -  ««») 

-f  v6  to  -  ^s)  4-  v?  (v?  -  %)  4-  vs  (vg  -  u4)  +  v9  (w9  -  Ws)  +  v10  (VM  -  ^0)- 


§  2-34.  TAe  limiting  process^.  We  assume  that  6  is  a  simply  connected 
region  in  the  ^-plane  with  a  boundary  F  formed  of  a  finite  number  of  arcs 
with  continuously  turning  tangents.  If  v  is  an  integrable  function  defined 
within  G  we  shall  use  the  symbol  0  {v}  to  denote  the  integral  of  v  over  the 
area  G  and  a  similar  notation  will  be  used  for  integrals  of  v  over  portions 
of  G  which  are  denoted  by  capital  letters. 

Let  Gh  be  the  lattice  region  associated  with  the  mesh-width  h  and  the 
region  G,  and  let  the  symbol  Gh  [v]  be  used  to  denote  the  sum  of  the 
values  of  v  over  the  lattice  points  of  G.  Also  let  the  symbol  I\  (v)  be  used 
for  the  sum  of  the  values  of  v  over  the  boundary  points  which  form  the 
boundary  I\  of  Gh.  This  notation  will  be  used  also  for  a  portion  of  Gh 
denoted  by  a  capital  letter  and  for  the  lattice  region  Qh*  belonging  to  a 
partial  region  Q*  of  G. 

Now  let  /  (x,  y)  be  a  given  function  which  is  continuous  (D,  2)  in  a 
region  enclosing  G  and  let  u  (#,  y)  be  the  solution  of  (A)  which  takes  the 
same  value  as  /  (#,  y)  at  the  boundary  points  of  Gh.  We  shall  prove  that 
as  h  ->  0  the  function  uh  (x,  y)  converges  towards  a  function  u  (xy  y)  which 

f  R.  Courant,  K.  Friedrichs  and  H.  Lewy,  Math.  Ann.  vol.  c,  p.  32  (1928).  See  also  J.  le 
Roux,  Journ.  de  Math.  (6),  vol.  x,  p.  189  (1914);  R»  G.  D.  Richardson,  Trans.  Amer.  Math.  Soc. 
vol.  xvm,  p.  489  (1917);  H.  B.  Phillips  and  N.  Wiener,  Journ.  Math,  and  Phys.  Mass.  Inst.  Tech. 
vol.  n,  p.  105  (1923). 

10-2 


148    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

satisfies  the  partial  differential  equation  V2u  =  0  and  takes  the  same  value 
as  /  (#,  y)  at  each  of  the  points  of  F.  We  shall  further  show  that  for  any 
region  lying  entirely  within  G  the  difference  quotients  of  uh  of  arbitrary 
order  tend  uniformly  towards  the  corresponding  partial  derivatives  of 
u  (x,  y). 

In  the  convergence  proof  it  is  convenient  to  replace  the  boundary 
condition  u  —  f  on  F  by  the  weaker  requirement  that 

<$>{(^~/)2}-*0     asr->0, 

where  Sr  is  that  strip  of  G  whose  points  are  at  a  distance  from  F  smaller 
than  r. 

The  convergence  proof  depends  on  the  fact  that  for  any  partial  region 
G*  lying  entirely  within  G,  the  function  uh  (x,  y)  and  each  of  its  difference 
quotients  remains  bounded  and  uniformly  continuous  as  h  ->  0,  where 
uniform  continuity  is  given  the  following  meaning  : 

There  is  for  any  of  these  functions  wh  (x,  y)  a  quantity  8  (e),  depending 
only  on  the  region  and  not  on  h,  such  that  if  wh(p)  denote  the  value  of  the 
function  at  the  point  P  we  have  the  inequality 

I  *VP)  -  *^(PI)  |  <  * 

whenever  the  two  lattice  points  P  and  Pl  of  the  lattice  region  Gh  lie  in  the 
same  partial  region  and  are  separated  by  a  distance  less  than  8  (e). 

As  soon  as  the  foregoing  type  of  uniform  continuity  has  been  established 
we  can  in  a  well-known  manner  f  select  from  our  functions  uh  a  partial 
sequence  of  functions  which  tend  uniformly  in  any  partial  region  6?* 
towards  a  limit  function  u  (x,  y}  while  the  difference  quotients  of  uh  tend 
uniformly  towards  that  of  u  (x,  y}  differential  coefficients.  The  limit  func- 
tion then  possesses  derivatives  of  order  n  in  any  partial  region  G*  of  G  and 
satisfies  V*u  =  0  in  this  region.  If  we  can  also  show  that  u  satisfies  the 
boundary  condition  we  can  regard  it  as  the  solution  of  our  boundary 
problem  for  the  region.  G.  Since  this  solution  is  uniquely  determined,  it 
appears  then  that  not  only  a  partial  sequence  of  the  functions  uh  but  this 
sequence  of  functions  itself  possesses  the  desired  convergence  property  as 
A-+0. 

The  uniform  continuity  of  our  quantities  may  be  established  by  proving 
the  following  lemmas  : 

(1)  As  h  -*  0  the  sums  h*Qh  [u*]  and  h*Gh  [ux2  +  uyz]  remain  bounded. 

(2)  If  w  =  wh  satisfies  the  difference  equation  (A)  at  a  lattice  point 
of  Gh  and  if,  as  h  ->  0  the  sum  h2G  A*  [w2]  ,  extended  over  a  lattice  region 
Gh*  associated  with  a  partial  region  G*  of  G,  remains  bounded,  then  for 
any  fixed  partial  region  6?**  lying  entirely  within  6?*  the  sum 


f  See  for  instance,  Kellogg's  Foundations  of  Potential  Theory,  p.  265.   The  theorem  to  be 
used  is  known  as  Ascoli's  theorem;  it  is  discussed  in  §  4-45. 


Inequalities  149 

over  the  lattice  region  6?A**  associated  with  6**,  likewise  remains  bounded 
as  Ti->  0.  When  this  is  combined  with  (1)  it  follows  that,  because  all  the 
difference  quotients  w  of  the  function  uh  also  satisfy  the  difference  equation 
(A),  each  of  the  sums  h2Gh*  [w2]  is  bounded. 

(3)  From  the  boundedness  of  these  sums  it  follows  that  the  difference 
quotients  themselves  are  bounded  and  uniformly  continuous  as  h  ->  0. 

The  proof  of  (1)  follows  from  the  fact  that  the  functional  values  uh  are 
themselves  bounded.  For  the  greatest  (or  least)  value  of  the  function  is 
assumed  on  the  boundary *f  and  so  tends  towards  a  prescribed  finite  value. 
The  boundedness  of  the  sum  h2Gh  [ux2  -f  uy2]  is  an  immediate  consequence 
of  the  minimum  property  of  our  lattice  function  which  gives  in  particular 

h*Gh  [ux2  +  uy2]  <  WQK  (fx2  +  A2], 

but  as  h->  0  the  sum  on  the  right  tends  to  G  \(  J  )   -f  (  ~    )  ,v,  which,  by 

[\dx/       \vy>) 

hypothesis,  exists. 

To  prove  (2)  we  consider  the  sum  h2Ql  [ivx2  -f  w^  +  wy2  -f-  w^2],  where 
the  summation  extends  over  all  the  interior  points  of  a  square  Qt.  Now 
Green's  formula  gives 

h2Qi  K2  +  ™**  +  Wy2  +  ?V]  =  2  (w2)  -  S  (w2), 

1  0 

where  £j  and  20  are  respectively  the  boundary  of  Qv  and  the  square 
boundary  of  the  lattice  points  lying  within  Sj . 

We  now  consider  a  series  of  concentric  squares  Q0 ,  Ql ,  ...  Q  v  with  the 
boundaries  S0,  S19  ...  2^v-  Applying  our  formula  to  each  of  these  squares 
and  observing  that  we  have  always 

2h2Q0  [wx2  +  wy2]  <  Ji2Qk  [w,2  -f  w22  +  wy2  +  wg*]         (k  >  1), 
we  obtain 

2h2Q0  [wx2  +  wy2]  <   2  (w2)  -  2  (iv2)         (0  <  k  <  n). 

A*  f  1  k 

Adding  n  inequalities  of  this  type  we  obtain 

2nh2Q0  [wx2  -{-  wy2]  <  S  (w2)  -  S  (w2)  <  S  (w2). 

n  0  n 

Summing  this  inequality  from  n  =  1  to  n  =  N  we  get 
N2h2Q0  [wx2  -f  ?V]  <  QN  [w2]. 

Diminishing  the  mesh-width  h  we  can  make  the  squares  Q0  and  QN 
converge  towards  two  fixed  squares  lying  within  0  and  having  corre- 
sponding sides  separated  by  a  distance  a.  In  this  process  Nh  ->  a  and  we 

have  independently  of  the  mesh-width 

/>2 
h2Q0[wx2^wv2]<a  QN[w2]. 

Uf 

t  On  account  of  equation  (A)  the  value  of  uh  at  an  internal  point  is  the  mean  of  the  values 
at  the  four  neighbouring  points  and  so  cannot  be  greater  than  all  of  them,  consequently  the  greatest 
value  of  M*  cannot  occur  at  an  internal  point. 


150    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

With  a  sufficiently  small  value  of  h  this  inequality  holds  with  another 
constant  a  for  any  two  partial  regions  of  G,  one  of  which  lies  entirely 
within  the  other.  Hence  the  surmise  in  (2)  is  proved. 

To  prove  that  uh  and  all  its  partial  difference  quotients  w  remain 
bounded  and  uniformly  continuous  as  h  ->  0  we  consider  a  rectangle  R 
with  corners  P0,  QQJ  P,  Q  and  with  sides  P0QQ,  PQ  which  are  x-linesf  of 
length  a.  Denoting  these  lines  by  the  symbols  XQ  ,  X  respectively  we  start 

from  the  formula          „          p       7  ir  /     v   ,    79r»  r       -, 
K^O  -  wp*  =  hX  (wx)  +  h*R  [wxy] 

and  the  inequality 

|  WQQ  _  WPQ  |  <  hX  (|  wx  |)  +  h*R  [|  wxy  |]  ......  (C) 

which  is  a  consequence  of  it.  We  now  let  X  vary  continuously  between  an 
initial  position  Xl  at  a  distance  6  from  XQ  and  a  final  position  X2  at  a 
distance  26  from  X0  and  sum  the  (b/h)  -f  1  inequalities  (C)  associated  with 
X'a  which  pass  through  lattice  points.  We  thus  obtain  the  inequality 


where  the  summations  on  the  right  are  extended  over  the  whole  rectangle 
P0Q0P2Q2.   By  Schwarz's  inequality  it  then  follows  that 

|  wp»  -  u#>  |  <  (2a/6)*  (A2#2  [wx2])*  -f  (2a6)i  (&2#2  [^xv2])*. 

Since,  by  hypothesis,  the  sums  which  occur  heVe  multiplied  by  A2 
remain  bounded,  it  follows  that  as  a  ->  0  the  difference  |  wp<>  —  wA  |  -^  0 
independently  of  the  mesh-  width,  since  for  each  partial  region  G*  of  6? 
the  quantity  b  can  be  held  fixed.  Consequently,  the  uniform  continuity  of 
w  =  wh  is  proved  for  the  ^-direction.  Similarly,  it  holds  for  the  t/-direction 
and  so  also  for  any  partial  region  (7*  of  G.  The  boundedness  of  the  function 
wh  in  G*  finally  follows  from  its  uniform  continuity  and  the  boundedness 
of  h*G*[wh*]. 

By  this  proof  we  establish  the  existence  of  a  partial  sequence  of 
functions  uh  which  converge  towards  a  limit  function  u  (x,  y)  and  are, 
indeed,  continuous,  together  with  all  their  difference  quotients,  in  the 
sense  already  explained,  for  each  inner  partial  region  of  G.  This  limit 
function  u  (x,  y)  is  thus  continuous  (Z>,  n)  throughout  (?,  where  n  is 
arbitrary,  tod  it  satisfies  the  potential  equation 


In  order  to  prove  that  the  solution  fulfils  the  boundary  condition 
formulated  above  we  shall  first  of  all  establish  the  inequality 

h2Sr,h  [v2]  <  ArWSfth  [vx2  -f  V]  +  Brhrh  (v2),       ......  (D) 

where  Sf)h  is  that  part  of  the  lattice  region  Gh  which  lies  within  the 

t  This  term  is  used  here  to  denote  lines  parallel  to  the  axis  of  x. 


Properties  of  the  Limit  Function  151 

boundary  strip  Sr,  which  is  bounded  by  F  and  another  curve  Fr.  The 
constants  A,  B  depend  only  on  the  region  and  not  on  the  function  v  or 
the  mesh-width  h. 

To  do  this  we  divide  the  boundary  F  of  G  into  a  finite  number  of  pieces 
for  which  the  angle  of  the  tangent  with  either  the  x  or  t/-axis  is  greater 
than  30°.  Let  y,  for  instance,  be  a  piece  of  F  which  is  sufficiently  steep 
(in  the  above  sense)  relative  to  the  #-axis.  The  x-lines  through  the  end- 
points  of  the  piece  y  cut  out  on  Fr  a  piece  yr  and  together  with  y  and  yr 
enclose  a  piece  sr  of  the  boundary  strip  Sr  .  We  use  the  symbol  sft  h  to  denote 
the  portion  of  Gh  contained  in  sr  and  denote  the  associated  portion  of  the 
boundary  FA  by  yh. 

We  now  imagine  an  o?-line  to  be  drawn  through  a  lattice  point  Ph  of 
Sr  th.  Let  it  meet  the  boundary  yh  in  a  point  Ph.  The  portion  of  this 
#-line  Xh  which  lies  in  sr  th  we  call  pr  th.  Its  length  is  certainly  smaller 
than  cr,  where  the  constant  c  depends  only  on  the  smallest  angle  of  in- 
clination of  a  tangent  of  y  to  the  #-axis.  __ 

Now  between  the  values  of  v  at  Ph  and  Ph  we  have  the  relation 
vph  ^  vph  ±  hxh  (vx). 

Squaring  both  sides  and  applying  Schwarz's  inequality,  we  obtain 
(vf*)*  <  2  (iA)2  +  2crhpTfh  (vx2). 

Summing  with  respect  to  Ph  in  the  ^-direction,  we  get 


Summing  again  in  the  ^/-direction  we  obtain  the  relation 
hsr,h  [v*]  <  2crTh  (v?*)*  +  2c*r*Sr,h  [v,*]. 

Writing  down  the  inequalities  associated  with  the  other  portions  of  F 
and  adding  all  the  inequalities  together  we  obtain  the  desired  inequality 
(D). 

By  similar  reasoning  we  can  also  establish  the  inequality 

h*Qh  0*]  <  Clhrh  (v*)  -f  c2h*Gh  (V  +  V] 

in  which  the  constants  cls  c2  depend  only  on  the  region  G  and  not  on  the 
mesh  division. 

We  now  put  vh  =  uh  —  fh  so  that  vh  =  0  on  FA  . 

Then,  since  h2Gh  [vx2  -f  vy2]  remains  bounded  as  h  ->  0,  we  obtain  from 

(D)  (h*/r)Srtk[v*]<Kr,  ......  (E) 

where  K  is  a  constant  which  does  not  depend  on  the  function  v  or  the  mesh- 
width.  Extending  the  sum  on  the  left  to  the  difference  S^h  —  Spfh  of  two 
boundary  strips,  the  inequality  (E)  still  holds  with  the  same  constant  K 
and  we  can  pass  to  the  limit  h  -*  0. 
From  the  inequality  (D)  we  then  get 

(l/r)S[v*]<Kr, 


152     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

where  8  —  Sr  —  Sfl  and  v  —  u  —  /.    Now  letting  />  ->  0,  we  obtain  the 
inequality  ^  ^  ^  ^  (1/jp)  ^  [(u  _  /)2]  ^^  ^ 

which  signifies  that  the  limit  function  satisfies  the  prescribed  boundary 
condition. 

§  2-41.  The  derivation  of  physical  equations  from  a  variational  principle. 
\  concise  expression  may  be  given  to  the  principles  from  which  an  equation 
or  set  of  equations  is  derived  by  using  the  ideas  of  the  "  Calculus  of 
Variations*."  This  expression  is  useful  for  several  purposes.  In  the  first 
place  a  few  methods  are  now  available  for  the  direct  solution  of  problems 
in  the  "Calculus  of  Variations"  and  these  can  sometimes  be  used  with 
advantage  when  the  differential  equations  are  hard  to  solve.  Secondly, 
when  an  integral's  first  variation  furnishes  the  desired  physical  equations 
the  expression  under  the  integral  sign  may  be  used  with  advantage  to 
obtain  a  transformation  of  the  physical  equations  to  a  new  set  of  co- 
ordinates, for  the  transformation  of  the  integral  is  generally  much  easier 
than  the  transformation  of  the  differential  equations  and  the  transformed 
equations  can  generally  be  derived  from  the  transformed  integral  by  the 
methods  of  the  "Calculus  of  Variations,"  that  is,  by  the  Eulerian  rule. 

To  illustrate  the  method  we  consider  the  variation  of  the  integral 

r   i  f  ff  [f3v\2   isv\2  /2F 

/==  6  a  'I   +    "a  -)   +  br 

2JJJ  [\dxj        \dy/        \dz 

when  the  dependent  variable  V  is  alone  varied  and  its  variation  is  chosen 
so  that  it  vanishes  on  the  boundary  of  the  region  of  integration.   We  have 


Now  by  a  fundamental  property  of  the  signs  of  variation  and  differentia- 
tion  a  I/      a 

^  0  V         0      ~J7 

8-x—  —  Q-  (SF),  etc. 

dx      8x  v      h 

Hence  87  -  f  I  f  \~  j^  (8V)  +  ...1  dxdydz 


V~dS-      \8V.V2Vdxdydz. 

dn  J J J  y 

The  surface  integral  vanishes  because  8F  =  0  on  the  boundary,  conse- 
quently the  first  variation  87  vanishes  altogether  if  F  satisfies  everywhere 

the  differential  equation  .-.^^ 

^  V2!/  =  0. 

The  condition  8  F  =  0  on  the  boundary  means  that  as  far  as  the  possible 
variations  of  F  are  concerned  F  is  specified  on  the  boundary.   It  is  easily 

*  The  reader  may  obtain  a  clear  grasp  of  the  fundamental  ideas  from  the  monograph  of 
G.  A.  Bliss,  "Calculus  of  Variations,"  The  Carus  Mathematical  Monographs  (1925). 


Variational  Principles  153 

seen  that  a  function  V  with  the  specified  boundary  values  gives  a  smaller 
value  of  /  when  it  is  a  solution  of  V2  V  =  0,  regular  within  the  region,  than 
if  it  is  any  other  regular  function  having  the  assigned  values  on  the 
boundary. 

In  the  foregoing  analysis  it  is  tacitly  assumed  that  V2  V  exists  and  is 
such  that  the  transformation  from  the  volume  integral  to  the  surface 
integral  is  valid.  If  V  is  assumed  to  be  continuous  (Z>,  2)  there  is  no  difficulty 
but,  as  Du  Bois-Reymond  pointed  out*,  it  is  not  evident  that  a  function  V 
which  makes  87  =  0  does  have  second  derivatives.  This  difficulty,  which 
has  been  emphasised  by  Hadamardf  and  LichtensteinJ,  has  been  partly 
overcome  by  the  work  of  Haar  §.  There  are  in  fact  some  sufficient  conditions 
which  indicate  when  the  derivation  of  the  differential  equation  of  a  varia- 
tion problem  is  permissible. 

For  the  corresponding  variation  problem  in  one  dimension  there  is 
a  very  simple  lemma  which  leads  immediately  to  the  desired  result.  The 
variation  problem  is 

8 


where  xl  and  x2  are  constants  and  8  V  is  supposed  to  be  zero  for  x  =  x\ 
and  for  x  —  x2  . 

dV 
Writing     -  =  M  ,  8  V  =  U  we  have  to  show  that  if 


-« 

for  all  admissible  functions  U  which  satisfy  the  conditions 

U(x,)=  U(x2)  =  0  ......  (B) 

then  M  is  a  constant  (Du  Bois-Reymond's  Lemma). 

To  prove  the  lemma  we  consider  the  particular  function 

U(x)  =  (x2  -  x)  \XM  (f)  dg  -  (x  -  x,)  PMtf)  dg, 

J  Xt  JX 

which  satisfies  (B)  and  gives  at  any  point  x  where  M  (x)  is  continuous 


-  c],  say. 

*  P.  du  Bois-Reymond,  Math.  Ann.  vol.  xv,  pp.  283,  564  (1879). 

f  J.  Hadamard,  Comptes  Rendus,  vol.  CXLIV,  p.  1092  (1907). 

J  L.  Lichtenstein,  Math  Ann.  vol.  LXEX,  p.  514  (1910). 

§  A.  Haar,  Journ.  fur  Math.  Bd.  CXLIX,  S.  1  (1919);  Szeged  Acta,  t.  in,  p.  224  (1927).  Haar 
shows  that  in  the  case  of  a  two-dimensional  variation  problem  the  equation  87  =  0  leads  to  a  pair 
of  simultaneous  equations  of  the  first  order  in  which  there  is  an  auxiliary  function  W  whose 
elimination  would  lead  to  the  Eulerian  differential  equation  if  the  necessary  differentiations 
could  be  performed.  Many  inferences  may,  however,  be  derived  directly  from  the  simultaneous 
equations  without  an  appeal  to  the  Eulerian  equation. 


154    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

We  shall  now  assume  that  M(x)  is  continuous  bit  by  bit  (piece wise  con- 
tinuous) so  that  this  equation  holds  in  the  interval  (xl ,  x2)  except  possibly 

at  a  finite  number  of  points.  The  functions  M  (x),  -j-  are  then  undoubtedly 

(tx 

integrable  over  the  range  (xl9  x2)  and,  on  account  of  the  end  conditions  (B) 
satisfied  by  U(x),  we  may  write  (A)  in  the  form 


With  the  value  adopted  for  U  this  equation  becomes 

cxt 

[M  (x)  —  c]2  dx  =  0, 

dM  d2V 

and  implies  that  M  (x)  —  c,  hence  -7-  =  0  and  -v-z  =  0. 

a#  dx2 

An  extension  of  this  analysis  to  the  three-dimensional  case  is  difficult. 
To  avoid  this  difficulty  it  is  customary*  to  limit  the  variation  problem  and 
to  consider  only  functions  that  are  continuous  (D,  2)  throughout  the  region 
of  integration.  The  function  V  and  the  comparison  function  V  +  U  are 
supposed  to  belong  to  the  field  of  functions  with  the  foregoing  property. 
The  problem  is  to  find,  if  possible,  a  function  V  of  the  field  such  that  81 
is  zero  whenever  U  belongs  to  the  field  and  is  zero  on  the  boundary  of  the 
region  of  integration. 

Even  when  the  problem  is  presented  in  this  restricted  form  a  lemma  is 
needed  to  show  that  V  necessarily  satisfies  the  differential  equation.  We 
have,  in  fact,  to  show  that  if 

U.V2V  dxdydz^  0, 

for  all  admissible  functions  U,  then  V2  V  =  0. 

The  nature  of  the  proof  may  be  made  clear  by  considering  the  one- 
dimensional  case.  We  then  have  the  equations 

<f)  (x)  U  (x)  dx  =  0, 

and  the  conditions : 

U  (x)  is  continuous  (Z>,  2),         <f>(x)  is  continuous  in  (xl9  x2). 
Since  U  (x)  is  otherwise  arbitrary  we  may  choose  the  particular  function 
U  (x)  =  (x  —  a)4  (b  —  #)4  xl  <  a  <  x  <  6  <  x2 

=  0  otherwise. 

If  <f>(x)  were  not  zero  throughout  the  interval  (xlf  x2)  it  would  have  a 
definite  sign  (positive,  say)  in  some  interval  (a,  6)  contained  within  (xl9  #2), 

*  See,  for  instance,  Hilbert-Courant,  Methoden  der  Mathematischen  Physik,  vol.  I,  p.  165. 


Fundamental  Lemmas  155 

but  this  is  impossible  because  with  the  above  form  of  U(x)  the  integral 
<f>  (x)  U(x)dx  is  positive. 

J X\  i 

To  extend  this  lemma  to  the  three-dimensional  problem  it  is  sufficient 
to  consider  a  function  U(x,  y,  z)  which  has  a  form  such  as 

within  a  small  cube  with  (alf  a2,  a3),  (blt  62>  ^3)  as  ends  of  a  diagonal,  the 
value  of  U  outside  the  cube  being  zero. 

In  this  way  it  can  be  shown  that  a  field  function  V  for  which  81  —  0 
is  necessarily  a  solution  of  V2  V  =  0.  The  foregoing  analysis  does  not  prove, 
however,  that  such  a  function  exists. 

Similar  analysis  may  be  used  to  derive  the  equation  V2<f>  +  k*(f>  =  0  from 
a  variational  principle  in  which 

8  If  \Ldxdydz  =  0, 


When  the  potential  <f>  is  of  the  form  -  cos  kr  the  volume  integral  is 
finite  although  the  integral  k2<f>2  is  not*. 

EXAMPLES 

i.  if  /ass  f/T(li)8" 

the  equation  87  =  0  may  be  satisfied  by  making  V  —  f(x  +  y)  +  g(x  —  y)  where  /  and  g 
have  first  derivatives  but  not  necessarily  second  derivatives.  [Hadamard.] 

2.     The  variation  problem 

8       F(VX,  Vy,  x,  y)dxdy  =  0 

leads  to  the  simultaneous  equations 

W-dF       W  -~™~ 
x~dVy'        v~     BVX' 

the  suffixes  x,  y  denoting  differentiations  with  respect  to  these  variables.  [A.  Haar.] 

§  2»42.  The  general  Eulerian  rule.  To  formulate  the  general  rule  for 
finding  the  equations  which  express  that  the  first  variation  of  an  integral 
is  zero  we  consider  the  variation  of  an  integral 

f  f      f 
/  =      ...    L  dxl  dx2  . . .  dxn , 

where  L  is  a  function  of  certain  quantities  and  their  derivatives.   For 

,     *  See,  for  instance,  the  remarks  made  by  J.  Lennard- Jones,  Proc.  London  Math.  Soc.  vol.  xx, 
p.  347  (1922). 


156    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

brevity  we  use  Suffixes  1,  2,  etc.  to  denote  derivatives  with  respect  to 
#!,  x2,  etc.  If  there  are  m  quantities  u,  v,  w,  ...  which  are  varied  inde- 
pendently except  for  certain  conditions  at  the  boundary  of  the  region  of 
integration,  there  are  m  Eulerian  equations  which  are  all  of  type 


o  =     _  -     -  -    -    ~  e 

du      S,idx3\duj      2la^it^i  dxadxt\dust)   at 
1    "     "    n          33        /  SL 

~  3  !  r?t  .?!  £  aavaar.a*;  l 

these  are  often  called  the  Euler-Lagrange  differential  equations,  but  for 
brevity  we  shall  call  them  simply  the  Eulerian  equations. 

If  /j  ,  12  ,  .  .  .  ln  are  the  direction  cosines  of  the  normal  to  an  element  of 
the  boundary,  the  boundary  conditions  are  of  types 


, 
' 


rdL    i     a  /    a/A     i       a»   /     a/A      i 
ldu.   2!2iaTAe"a^J  +  3!2ja*;aa:Ae"'9«rJ    "T 


r-l 


0=    S 

a-! 

There  are  m  boundary  conditions  of  the  first  type,  mn  boundary  con- 
ditions of  the  second  type,  ^mn(n—  1)  boundary  conditions  of  the  third 
type,  and  so  on.  In  these  equations  the  coefficients  $st,  €rst  are  constants 
which  are  defined  as  follows  : 

€8t  -    1  S  ^   t, 

=  2  8=  t, 

erst  =  1  r  ^  s  ^  t, 
=  2  r  =  s  ^  t, 
=  6  r  =  s  ^  t. 

The  equations  (A)  are  obtained  by  subjecting  the  integral  to  repeated 
integrations  by  parts  until  one  part  of  the  integral  is  an  integral  over  the 
boundary  and  the  other  part  is  of  type 

||...  l[USu+  V8v+  WSw+  ...]dx,  ...  dxn. 
The  equations  ^  ^     y  _  ^     w  =  Q? 

are  then  the  Eulerian  differential  equations*,  while  the  boundary  integral  is 

(dS  [USu  +  S  UtSut  +  L  UnSurt  +  ...], 


and  the  boundary  conditions  are 

f/  =  0,     f/(=0     («=1,  2,...),     f/rt=0     (r=  1,2,  ...;<=  1,2,...). 

*  For  general  properties  of  the  Eulerian  equations  see  Ex.  2,  p.  183,  and  the  remarks  at  the 
end  of  the  chapter. 


The  Eulerian  Equations  157 

Typical  integrations  by  parts  are 

^  d    [su    —  1  -ou     a  (SL\ 

a  r      ail    j)  r,    l_/^\]     s     ^2    /^\ 

a2    /dL 


rs    3L]     a  ra    a  /3L\i     ,      a 

°UI  i         \  ~  ^        \  °u   ^  -    \  ^        }\   +  OU  ~ 

[_       ^Uu\      vxi  L       dx2\duu/ \  dx^ 


~     a£  __  a  r\     3//1     a  r      a  /a//\~]    ~    d*  /dL' 

The  reason  for  the  introduction  of  the  factors  tst  is  now  apparent. 

When  L  depends  only  on  a  single  quantity  u  and  its  first  derivatives 
the  Eulerian  equation  is  of  the  second  order.  The  variation  problem  is 
then  said  to  be  regular  when  this  partial  differential  equation  is  of  elliptic 
type.  The  distinction  between  regular  and  irregular  variation  problems 
becomes  apparent  when  terms  involving  the  square  of  8u  are  retained  and 
the  sign  of  the  sum  of  these  terms  is  investigated  (Legendre's  rule). 

When  a  variation  problem  is  irregular  it  is  not  certain  that  the  boundary 
conditions  suggested  by  the  variation  pioblem  will  be  equivalent  to  those 
which  are  indicated  by  physical  considerations. 

For  a  physically  correct  variation  problem  a  direct  method  of  solution 
is  often  advantageous.  The  well-known  method  of  Rayleigh  and  Ritz 
is  essentially  a  method  of  approximation  in  which  the  unknown  function 
is  approximated  by  a  finite  series  of  functions,  each  of  which  satisfies  the 
specified  boundary  conditions.  The  coefficients  in  the  series  are  chosen 
so  as  to  make  81  =  0  when  each  coefficient  is  varied.  The  problem  is  thus 
reduced  to  an  algebraic  problem. 

§  2-431.  The  transformation  of  physical  equations.  In  searching  for 
simple  solutions  of  the  partial  differential  equations  of  physics  it  is  often 
useful  to  transform  the  equations  to  a  new  set  of  co-ordinates  and  to  look 
for  solutions  which  are  simple  functions  of  these  co-ordinates.  The  necessary 
transformations  can  be  made  without  difficulty  by  the  rules  of  tensor 
analysis  and  the  absolute  calculus,  but  sometimes  they  may  be  obtained 
very  conveniently  by  transforming  to  the  new  co-ordinates  the  integral 
which  occurs  in  a  variational  problem  from  which  they  are  derived.  The 
principle  which  is  used  here  is  that  the  Eulerian  equations  which  are 
derived  from  the  transformed  integral  must  be  equivalent  to  the  Eulerian 
equations  which  were  derived  from  the  original  integral  because  each  set 
of  equations  means  the  same  thing,  namely,  that  the  first  variation  of  the 
integral  is  zero.  A  formal  proof  of  the  general  theorem  of  the  covariance 
of  the  Eulerian  equations  can,  of  course,  be  given  *,  but  in  this  book  we  shall 

*  L.   Koschmieder,   Math.  Zeits.  Bd.  xxiv,  S.    181,  Bd.  xxv,  S.   74  (1926);    Hilbert  and 
Courant,  I.e.  p.  193. 


158    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

regard  this  property  of  covariance  as  a  postulate.  It  is  well  known,  of 
course,  that  the  postulate  leads  to  the  Lagrangian  equations  of  motion  in 
the  simple  case  when  the  integral  is  of  type 

Ldt, 

where  L  =  /  [qly  q2,  ...  qn\  qlyq2,  ...  qn] 

=  T  -  F, 

the  Lagrangian  equations  being  of  type 

dL     c 


0  =   ,  __.   , 

dq3      dt  \SqsJ ' 

The  quantity  T  here  denotes  the  kinetic  energy  and  V  the  potential 
energy.  V  is  a  function  of  the  co-ordinates  which  specify  a  configuration 
of  the  dynamical  system,  while  T  is  a  positive  quantity  which  depends  on 
both  the  q's  and  their  rates  of  change,  which  are*  denoted  here  by  g's. 

In  the  simple  case  when 

n   n 

m  IVV/v    A  A 

j-    =  £  2j  Zj  drs  qr  qs , 

i  i 

n  n 
TT  l.VV/»     n  n 

11 

where  the  coefficients  ars ,  crs  are  constants,  the  covariance  of  the  equations 


is  easily  confirmed  by  considering  a  linear  transformation  of  type 

Ql  ^ 


Qn  =  Inl9l  +    •••  Inn9n> 

in  which  the  coefficients  lrs  are  constants. 

The  advantage  of  making  a  transformation  is  well  illustrated  by  this 
case,  because  when  the  transformation  is  chosen  so  that  the  expressions 
for  T  and  V  take  the  forms 


respectively,  the  Eulerian  equations  are  simply 

A,Q,  +  CSQ,  =  0, 
and  indicate  that  there  are  solutions  of  type 

Q.  =  as  cos  (n,t  +  ft),     (n*A,  =  Ct) 

where  as  and  j8,  are  arbitrary  constants.  These  co-ordinates  Q3  are  called 
the  normal  co-ordinates  for  the  dynamical  problem. 

Our  object  now  is  to  see  if  there  are  corresponding  sets  of  co-ordinates 
associated  with  a  partial  differential  equation. 


Transformation  of  Eulerian  Equations  159 

§  2-432.   To  transform  Laplace's  equation  to  new  co-ordinates  £,  77,  £ 
such  that 


dx*  +  dy2  +  dz2  =  ad^2  +  bd^  +  cdt* 

where  a,  6,  c,  /,  g,  h,  are  functions  of  £  ,  77,  £,  we  use  suffixes  a:,  t/,  z  to 
denote  differentiations  with  respect  to  x,  y,  z  and  suffixes  1,  2,  3  to  denote 
differentiations  with  respect  to  f  ,  rj,  £.  We  then  have 


7  df  d^dt,  say, 
.  '?»  t)  «/ 


say, 

h  -  af)  =  J*F,  say. 
Therefore 

2  F3  +  20V3  F,  +  2H  Vl  F2] 


By  Euler's  rule  87  =  0  when 

9  /  3L\      9_  /  9L_\      8  /  3L  \  _ 

a?  va  v\  )  +  ^  V3  F2y  +  a^  va  F3  )  ~  u> 

The  new  form  of  Laplace's  equation  is  thus 
nv     a 

UV  =  ^. 


^A      I     •*-*    n         i     -*•      oy 

c/f  C/TJ  c/4 

8 


If  the  original  integral  is 

P,a  +  FV2  +  Vz2-  XV*]dxdydz,  (A) 

the  transformed  integral  is 

where  U  =  L  —  |AF2/J, 

and  so  the  equation  V2F  +  AF  =  0, 

which  is  derived  from  (A)  by  Euler's  rule,  transforms  into  the  equation 

DV  +  XV IJ  =  0 


160    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 
which  is  derived  from  (B)  by  Euler's  rule.  This  shows  that  V2F  transforms 

into  J.DV, 

where  J2    a     h     g     =  1. 

!  &     b    f  \ 

\  9    f     c 

This  result  was  given  by  Jacobi*  with   the   foregoing   derivation.  The 
particular  case  in  which 


was  worked  out  by  Lam6.  The  result  is  that 

2  dV\      3  /  A8  8F 


This  result  is  of  great  importance  and  will  be  used  in  the  succeeding 
chapters  to  find  potential  functions  and  wave-functions  which  are  simple 
functions  of  polar  co-ordinates,  cylindrical  co-ordinates  and  other  co- 
ordinates which  form  an  orthogonal  system. 
In  the  special  case  when 

dx2  -f  dy*  +  dz*  -  *2  (r/£2  H-  d^  -f  d£2), 
Laplace's  equation  becomes 

a  /  dv\     a  /  dv\     a  /  sv\ 

^^\K    '^7    }^   ^     \K  -*-    H~   ^  K 

3|  V      9^  ;       Srj\     drjj       S 
and  implies  that  /c-F  is  a  solution  of 


if  K^  is  a  solution  of  this  equation. 

Inversion  is  one  transformation  which  satisfies  the  requirements,  for  in 
this  case  *  =  £/**,     y  = 


The  inference  is  that  if  F  (x,  y,  z)  is  a  solution  of  Laplace's  equation, 
the  functiont  , 

1F  x    y    z 

r      \r2'     r2'      r 
is  also  a  solution.  Another  transformation  which  satisfies  the  requirements 

is  ,  _     ax_  r2  -  a2  ,  _     r2  -f  a2 

^  ~  y~+Tz  '     ^  ~  2  (y"+  "  w)  '      Z  ~  2i  (y  +  iz)  ' 

*  Journ.  ftir  Math.  vol.  xxxvi,  p.  113  (1848).  See  also  J.  Larmor,  Caw6.  Phil.  Trans,  vol. 
xii,  p.  455  (1884);  vol.  xiv,  p.  128  (1886).  H.  Hilton,  Proc.  London  Math.  Soc.  (2),  vol.  xix, 
Records  of  Proceedings,  vii  (1921).  Some  very  general  transformation  formulae  are  given  by 
V.  Volterra,  Rend.  Lined,  ser.  4,  vol.  v,  pp.  599,  630  (1889). 

f  This  result  was  given  by  Lord  Kelvin  in  1845. 


Special  Transformations  161 

In  this  case 

dx'*  +  dy'*  +  dz'*  =  -  ^  --2  [dx*  +  <fy2 

~ 


a2   1 
w)  J 


and  we  have  the  result  that  if  F  (x,  y,  z)  is  a  solution  of  Laplace's  equation, 

ax  r2j--_a2  r2  + 

,-+-£  >     2  (*T-Mz)  '      2i 
is  also  a  solution. 

These  two  results  may  be  extended  to  Laplace's  equation  in  a  space  of 
n  dimensions  ^y     yy  ^y 

dX}2     dx22       '"  dxn2 
If  F  (#!,  #2,  ...  xn)  is  a  solution  of  this  equation,  and  if 


is  also  a  solution*,  and 

(*+ix\~*F\      T*-a2  r2t«8  ^3_  ^       1 

1  x  "^     2;  L2  (»i  +  ^2)  '     2t  (ij  +  tij  J      *!  Hh  is,  '  '  '  '  ^  +  ix  J 

is  a  second  solution.   We  shall  now  use  this  to  obtain  Brill's  theorem. 
Putting  Xj  4-  ix2  =  t,  x±  —  ix2  =  s,  the  differential  equation  becomes 


and  the  result  is  that  if  H2  =  a;32  4-  ...  xn2,  and  if 

jP  (s,t,x3,  ...  arn) 
is  a  solution,  then 


n 

~ 


is  also  a  solution.   Now  a  particular  solution  is  given  by 

8 

v  =  e        u  (t)  x$,  XD  ...  xn)) 
where  U  (t,  #3,  #4,  ...  xn)  is  a  solution  of  the  equation  of  heat  conduction 

dU         [d2U     d2U  d2 


which  is  suitable  for  a  space  of  n  —  2  dimensions.  The  inference  is  that  if 
U  is  one  solution  of  this  equation,  the  function 


a*      ax3        ax, 
fy     T"'*"~7 

*  The  first  result  is  given  by  B6cher,  Bull.  Amer.  Math.  Soc.  vol.  ix,  p.  459  (1903). 

B  II 


162    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

is  a  second  solution.    When  U  is  a  constant  the  theorem  gives  us  the 
particular  solution 


which  may  be  regarded  as  fundamental. 

EXAMPLES 

1.  Prove  that  if 

o  -  z  -  ct,        p**x  +  iyt       a  «~z  +  ct,       b  **  x  -  iy, 
a'-z'  +  ctf,     p-x'  +  iy',    a'  =  z'-ct',    b'  -  x'  -  iy  , 
the  relations        a'  (la  —  up  —  p)  =  —  na  4-  wp  4-  r  -f  j3'  (  —  ma  4-  v  j3  4-  g), 
a'  (wz  -f  ft  -  e)  =  -  tm  -  rib  -f  0  -  0'  (va  4-  mb  -  /), 
a'  (~  ma  +  v)8  -f  q)  -  foz  -f  j£  -I-  A;  -  6'  (Za  ~  w/3  -  ^p), 
a'  (va  -f-  mb  -  f)  =  ja  -  M  -f  «  +  6'  (wa  +  Z6  -  e), 

in  which  J,  m,  n,  u,  vt  w,f,  g,  h,  p,  q,  r,  s,  e,j,  k  are  arbitrary  constants,  lead  to  a  relation  of  type 
dxf*  +  dy'*  -h  dz'*  -  czdt'z  »  A2  (dx*  -f-  dy*  -f  ^22  - 

2.  Prove  that  if  z  -  c*  =  ^  +  (x  +  iy)  0, 

(z  -f  d)  0  =  *  -  (x  -  iy), 

the  relations  of  Ex.  1  give          z'  —  ct'  —  $  +  0'  (#'  -f-  iy'), 

0'  (*'  +  <*')  =  !/>'  -(*'-»/), 
where  X^  ^  w®'  ~~  v<t>'  +  w^'  4-  J, 


§2*51.  jPAe  equations  for  the  equilibrium  of  an  isotropic  elastic  solid. 
Let  u,  v.  w  be  the  components  of  the  small  displacement  of  a  particle, 
originally  at  x,  y,  z,  when  a  solid  body  is  sliglitly  deformed,  and  let  X  ,  Y,  Z 
be  the  components  of  the  body  force  per  unit  mass.  We  consider  the 
variation  of  the  integral 

/  =        Ldxdydz, 

where  L  =  S  —  W,  with 


vY 
25  -  (A  +  2/i)  (ux  +  vv  4-  wz)*  -f  ^  [K  + 


'A  and  p  being  positive  constants.  The  quantity  8  may  be  regarded  as  the 
strain  energy  per  unit  volume,  while  W  is  the  work  done  by  the  body 
forces  per  unit  volume.  The  densitj7  />  is  supposed  to  be  constant. 

We  now  wish  L  to  be  a  minimum  subject  to  the  condition  that  the 


Isotropic  Elastic  Solid  163 

values  of  u,  v  and  w  are  specified  at  the  boundary  of  the  solid.  The  Eulerian 
equations  of  the  "Calculus  of  Variations"  give 


dz 


where  Xx  =  2pux  +  X(ux  +  vy  +  wz),  Yz  =  Zy  =  ^  (ivy  4-  vg), 
Yy  =  2fjLVv  4-  A  (ux  4-  vv  4-  w,)9  Zx  =  .Yz  =  ^  (uz  +  wx)> 
Zz  =  2/xtik  4-  A  (ux  4-  vv  +  ^2),  Xv  =  Yx  =  /x  (t^  +  wy). 

The  quantities  JT,,.,  yy,  Zz,  ya,  Z^,  Xv,  are  called  the  six  components 
of  stress,  and  the  quantities 

exx*=ux)     evv=vv,     ezz=wz, 
eyz  =  wy  +  vz,     eza.  =  ft,  4-  w;x,     e^y  ==  vx  4-  uV9 

are  called  the  six  components  of  strain.    In  terms  of  these  quantities  28 
may  be  expressed  in  the  form 

28  =  Xxexx  4-  Yyevv  4-  Zzc«  +  r,ev,  4-  Zxegx  4-  ^e^, 
while  the  relations  between  the  components  of  stress  and  strain  are 
Xx  =  2fiexx  4-  AA,     Yz  =  Zy  =  Meyz, 
7V  -  2Metfv  4-  AA,     ZX  =  XZ  =  fieZX9 
Zz  =  2/iC«  4-  AA,     JCV  =  7X  =  nexy, 
A  =  ux-\-  vy  +  wz  =  exx  +  eyy  4-  ezz  . 
The  relations  may  also  be  written  in  the  form 
Eexx  =  Xx-  v(Yy+  Zz), 
Eeyy  =  Yv  -  a  (Zz  +  JQ, 
1?€M  =   Z,-^  cr  (Xx  +  Yv). 

The  coelSicient  E  is  Young's  modulus,  the  number  a  is  Poisson's  ratio, 
and  /A  is  the  modulus  of  rigidity.  The  quantity  A  is  the  dilatation  and 
—  A  the  cubical  compression.  When 

xm  =  YV  =  zz  =  -  ,,    y2  =  zx  =  xy  =  o, 

we  have  exx  =  eyy  -  ezz  =  -  jp/(3A  4-  2/i), 


-  A  =  !>/(A  + 
hence  the  quantity  k  defined  by  the  equation 

k  =  A  + 


164     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

is  called  the  modulus  of  compression.  The  different  elastic  constants  are 
connected  by  the  equations 

F_M?\+2M)  A  *__J0_ 

A  4-  /*      '  2~(A  +  /*)'  3  -  6(7* 

On  account  of  the  equations  of  equilibrium  the  expression  for  87  may 
be  written  in  the  form 

(XxSu  +  YxSv  +  ZxSw)  +  |  (Xy8u  +  Yybv  +  ZySw) 

r)  1 

4-  Sz  (Xz8u  4-  Yz8v  +  Z38w)    dxdydz, 
and  may  be  transformed  into  the  surface  integral 

[XvSu  -f  Yv 
where  Xv  =  IX  x  -f- 


4-  ^2,  . 

The  quantities  Xv,  YV9  Zv  are  called  the  components  of  the  surface 
traction  across  the  tangent  plane  to  the  surface  at  a  point  under  con- 
sideration. In  many  problems  of  the  equilibrium  of  an  elastic  solid  these 
quantities  are  specified  and  the  expressions  for  the  displacements  are  to  be 
found. 

The  equations  of  motion  of  an  elastic  solid  may  l3e  obtained  by  re- 

d'^ii         d^i)        v^ijo 
garding  -   ^2-  ,  —  -^  ,  —  ~  2   as  the  components  of  an  additional  body 

force  per  unit  mass.  The  equations  are  thus  of  type 

dXx     dX       3XZ  d2u 


§  2-52.    The  equations  of  motion  of  an  inviscid  fluid.  Let  us  consider  the 
variation  of  the  integral 

/  =  I  j  \\Ldxdydzdt, 
where  L  =  pl$  +  a-+l  (u*  -f  v»  +  w*)l  +/(/>), 


<>  <  , 

and  tt=^  +  a3S     v=?  +  a^,     w=-J-  +  a-£  .......  (A) 

ox         ox  cy         dy  oz         oz  ^    ' 

Varying  the  quantities  </»,  a,  j3  and  p  in  such  a  manner  that  the  variations 


Vortex  Motion  165 

of  </>  and  /?  vanish  on  a  boundary  of  the  region  of  integration  wherever 
particles  of  fluid  cross  this  boundary,  the  Eulerian  equations  give 


-  0 

~"' 


,  d    d       d    '    9        9 

where  -  -  ==  ^-  +  u  2-  +  v  =--  +  ti;  ^  . 

a£      d<          ox         ofy          dz 

If  p  =  />/'  (p)  —  /  (p)  it  is  readily  seen  that 

d^  __       I  dp       dv  __       I  dp       dw  __       1  3p 
~dt=z~~f>dx'     dt  =  ~~'pdy'      dt=~pdz' 

where  f  dp  +  ||  +  «  |£  +  J  (^  +  &  +  l^2)  =  j^  ^  .......  (E) 

If  £>  is  interpreted  as  the  pressure,  the  last  equation  is  the  usual  pressure 
equation  of  hydrodynamics  for  the  case  when  there  are  no  body  forces 
acting.  The  quantities  u,  v,  w  are  the  component  velocities  and  p  is  the 
density  of  the  fluid  at  the  point  x,  y,  z.  The  equation  (B)  is  the  equation 
of  continuity  and  the  equations  (D)  the  dynamical  equations  of  motion. 
The  relation  p  =  />/'  (p)  —  f  (p)  implies  that  the  fluid  is  a  so-called  baro- 
tropic  fluid  in  which  the  density  is  a  function  of  the  pressure.  It  should 
be  noticed  that  with  this  expression  for  the  pressure  the  formula  for  L 

becomes  T       ^  /A. 

L  =  F  (t)  -  p 

when  use  is  made  of  the  relation  (E). 

The  foregoing  analysis  is  an  extension  of  that  given  by  Clebsch*.  The 
fact  that  L  is  closely  related  to  the  expression  for  the  pressure  recalls  to 
memory  some  remarks  made  by  R.  Hargreaves|  in  his  paper  "A  pressure 
integral  as  a  kinetic  potential."  The  equations  of  hydrodynamics  may  also 
be  obtained  by  writing 

t+«d/t-l(u2+v2  +  ">*)}+  f(p)> 

and  varying  <f>,  a,  j3,  u,  v,  w  and  p  independently. 

The  equations  (A)  are  then  obtained  by  considering  the  variations  of 
uy  v  and  w.  These  equations  give  the  following  expressions  for  the  com- 

ponents of  vorticity  :  ~        ~       ~  ,     0. 

.  _  9^     9v  __  9  (a,  j8) 

f~  9i/~9z"9(*/,z)' 
_  du  _  dw  _  9  (a,  j8) 
**  ~  97  "  fa  ~~  d(zyx)  ' 
dv     du     9  (a,  j8) 


Crette'a  Journ.  vol.  LVI  (1859).  t  P^iL  Mag.  vol.  xvr,  p.  436  (1908). 


166     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

These  equations  indicate  that  a  =  constant,  j9  =  constant  are  the 
equations  of  a  vortex  line.  Now  the  equations  (C)  tell  us  that  a  and  )3 
remain  constant  during  the  motion  of  a  particle  of  fluid,  consequently  a 
vortex  line  moves  with  the  fluid  and  always  contains  the  same  particles. 

It  should  be  noticed  that  in  these  variational  problems  no  restrictions 
need  be  imposed  on  the  small  variations  80,  8/3  at  a  boundary  which  is 
not  crossed  by  particles  of  the  fluid  because  the  integrated  terms,  derived 
by  the  integration  by  parts,  vanish  automatically  at  such  a  boundary  of 
the  region  of  integration  on  account  of  the  equation  which  expresses  that 
fluid  particles  once  on  the  boundary  remain  on  the  boundary. 

§  2-53.  The  equations  of  vortex  motion  and  Liouville's  equation.  Let  us 
consider  the  variation  of  the  integral 

(A) 


'  o  x  '  ' 

O    "  >          "    n   /  "  \  » 

y  fa          d(x,y) 

the  expressions  for  u  and  v  being  chosen  so  as  to  satisfy  the  equation  of 
continuity,  *«     to  =  Q 

dx     dy         J 

for  the  two-dimensional  motion  of  an  incompressible  fluid. 

Varying  the  integral  by  giving  0  and  s  arbitrary  variations  which  vanish 
at  the  boundary  of  the  region  of  integration,  we  obtain  the  two  equations 


dxz     dy2      dx  \    dy)      dy  \    dx/ 
The  first  of  these  gives  s  =  g  (</r), 

where  g  (iff)  is  an  arbitrary  function  which,  when  the  region  of  integration 
extends  to  infinity,  must  be  such  that  the  integral  /  has  a  meaning.  This 
requirement  usually  means  that  u,  v  and  s  must  vanish  at  infinity.  With 
the  foregoing  expression  for  s  the  second  equation  takes  the  form 

£t  +  !^+?W?'M==°>  (B) 

which  is  no  other  than  Lagrange's  fundamental  equation  for  two-dimen- 
sional steady  vortex  motion.  In  the  special  case  when  g  ($)  =  Ae**,  where 
A  and  h  are  constants,  the  equation  becomes 


Liouville's  Equation  167 

This  equation,  which  also  occurs  in  Richardson's  theory  ot  the  space 
charge  of  electricity  round  a  glowing  wire*,  has  been  solved  by  Liouvillef, 
the  complete  solution  being  given  by 


where  a  and  r  are  real  functions  of  x  and  y  defined  by  the  equation 
a  +  ir  =  F  (x  +  iy)  and  F  (z)  is  an  arbitrary  function. 

Special  forms  of  F  which  lead  to  useful  results  have  been  found  by 
G.  W.  WalkerJ.   In  particular,  if  r*  =  x2  +  y2,  there  is  a  solution  of  type 

-**  —  Mr  I/TV    f?Vl  -  - 

and  when  n  =  1  the  component  velocities  are  given  by  the  expressions 

2i/  2x  ,„. 


, 

A  =  2/ahy  s  =  f-r  -  -  -  , 

1     '  h(a*  +  r2)  ' 

which  are  very  like  those  for  a  line  vortex  but  have  the  advantage  that 
they  do  not  become  infinite  at  the  origin.  If  we  write 

ds         ds         ds 

8  ~  -j*  —  u  a~  +  v  5~  > 
dt        dx        dy 

the  quantity  s  may  be  defined  by  the  equation 

s  =  —  a  tan"1  (y/x), 

and  has  a  simple  geometrical  meaning.  The  quantity  s  may  also  be  inter- 
preted as  the  velocity  of  an  associated  point  on  the  circle  r  =  a  which  is 
the  locus  of  points  at  which  the  velocity  is  a  maximum. 

It  should  be  noticed  that  if  we  use  the  variational  principle 

3  f  \(u2  +  v2  -  ^2)  dxdy  =  0,  ......  (D) 

the  corresponding  equation  is 

.  ......  <E> 


and  the  solution  corresponding  to  (C)  is  of  type 

~  __    2y  _  „__??__ 

U  ""  ""  ^(02^72)  '     V  "  A^a^-T2)  ' 

This  gives  an  infinite  velocity  on  the  circle  r  =  a. 

*  0.  W.  Richardson,  The  Emission  of  Electricity  from  hot  bodies,  Longmans  (1921),  p.  50.  The 
differential  equation  was  formulated  explicitly  by  ]Vf.  v.  La  lie,  Jahrbuch  d.  Radioaktivitat  u. 
Elektronik,  vol.  xv,  pp.  205,  257  (1918). 

f  LiouviUe's  Journal,  vol.  xvm,  p.  71  (1853). 

{  G.  W.  Walker,  Proc.  Roy.  Soc.  London,  vol.  xci,  p.  410  (1915);  BoUzmann  Festschrift,  p.  242 
(1904). 


168    Applications  of  the  Integral  Theorews  of  Gauss  and  Stokes 

Other  solutions  of  (B)  which  give  infinite  velocities  have  been  discussed 
by  Brodetsky  *.  It  seems  that  the  variational  principle  (A)  may  have  the 
advantage  over  (D)  in  giving  solutions  of  greater  physical  interest.  It 
should  be  noticed  that  if  a  boundary  of  the  region  of  integration  is  a  stream- 
line */r  =  constant,  it  is  not  necessary  for  8s  to  be  zero  on  this  boundary. 

When  the  motion  is  in  three  dimensions  an  appropriate  variation 
principle  is  87  =  0,  where 


/  =  £  n\(u*+v*  +  w*±$2)  dxdydz, 

and  the  upper  or  lower  sign  is  chosen  according  as  the  vortex  motion  is  of 
the  first  or  second  type.  To  satisfy  the  equation  of  continuity  when  the 
fluid  is  incompressible  and  the  density  uniform,  we  may  put 

11  =  a  (<J'  r)      »  =  9  ((7'  r)      m  =  d  (°>  r)       4  =  3  (*'  a>  r) 
8(»,2)'  d(z,x)'     W      d(x,y)'  d(x,y,zy 

A  set  of  equations  of  motion  is  now  obtained  by  varying  cr,  r  and  s  in 
such  a  way  that  their  variations  vanish  on  the  boundary  of  the  region  of 
integration.  These  equations  are 

3_(5,CT,  T)  =   () 

d  (x,  y,  z) 
,  Scr          da          da  _       9  («$,  «s,  cr) 

' 


and  are  equivalent  to  the  equations 
d  d(s>^ 


which  imply  that 


These  equations  give 

du  _       1  3  jp       dv  _       1  9j?       rfit?  _       1  dp 
dt  ~~      p  dx  '      dt~~      pdy*      dt  ~~      pdz* 

where  the  pressure  y>  is  given  by  the  equation 

^  +  \  (u2  -f  v2  +  w2  ±  s2)  =  constant. 

The  equation  of  continuity  may  also  be  derived  from  the  variation 
problem  by  adopting  Lagrange's  method  of  the  variable  multiplier.    In 

*  S.  Brodetsky,  Proceedings  of  the  International  Congress  for  Applied  Mechanics,  p.  374  (Delft, 
1924). 


Equilibrium  of  a  Soap  Film  169 

^\       ^\      ^\ 
this  method  /  is  modified  by  adding  A(  y  +5-^4-  ^   )  to  the  quantity 

within  brackets  in  the  integrand.  The  quantities  A,  u,  v,  w  are  then  varied 
independently.  It  is  better,  however,  to  further  modify  /  by  an  integration 
by  parts  of  the  added  terms.  The  variation  problem  then  reduces  to  the 
type  already  considered  in  §  2-52. 

§  2*54.  The  equilibrium  of  a  soap  film.  The  equilibrium  of  a  soap  film 
will  be  discussed  here  on  the  hypothesis  that  there  is  a  certain  type  of 
surface  energy  of  mechanical  type  associated  with  each  element  of  the 
surface.  This  energy  will  be  called  the  tension-energy  and  will  be  repre- 
sented by  the  integral 


!J 


TdS 


taken  over  the  portion  of  surface  under  consideration,  T  being  a  constant, 
called  the  surface  tension.  This  constant  is  not  dependent  in  any  way  upon 
the  shape  and  size  of  the  film  but  it  does  depend  upon  the  temperature. 
It  should  be  emphasised  that  a  soap  film  must  be  considered  as  having  .two 
surfaces  which  are  endowed  with  tension-energy.  The  tension-energy  is  not, 
moreover,  the  only  type  of  surface  energy;  perhaps  it  would  be  better  to 
say  film  energy ;  for  there  is  also  a  type  of  thermal  energy  associated  with 
the  film,  and  from  the  thermodynamical  point  of  view  it  is  generally 
necessary  to  consider  the  changes  of  both  mechanical  and  thermal  energy 
when  the  film  is  stretched. 

For  mechanical  purposes,  however,  useful  results  can  be  obtained  by 
using  the  hypothesis  that  when  a  film  stretched  across  a  hole  or  attached 
to  a  wire  is  in  equilibrium  under  the  forces  of  tension  alone,  the  total 
tension-energy  is  a  minimum. 

Assuming,  then,  as  our  expression  for  the  total  tension -energy  E 

E  =  2T  [  f  (1  +  zx2  4-  z,2)*  dxdy, 


the  z-co-ordinate  of  a  point  on  the  surface  or  rim  being  regarded  as  a 
function  of  x  and  y,  the  Eulerian  equation  of  the  Calculus  of  Variations 
gives  ~  ~ 


This  is  the  differential  equation  of  a  minimal  surface. 

When  the  film  is  subject  to  a  difference  of  pressure  on  the  two  sides 
and  the  fluid  on  one  side  of  the  film  is  in  a  closed  vessel  whose  pressure  is 
pl  while  the  pressure  on  the  other  side  of  the  film  is  p2  ,  there  is  pressure- 
energy  (pl  ~  p2)  V  associated  with  the  vessel  closed  by  the  film,  where  V 
is  the  volume  of  this  vessel.  Writing  V  in  the  form 


170    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

where  F0  is  a  constant  and  w  is  the  perpendicular  from  the  origin  to  the 
surface  element  dS,  we  consider  the  variation  of  the  integral 


Now  wH  =  z  —  xzx  —  yzy, 

and  so  the  differential  equation  of  the  problem  is 

0  =  ft  -  P*  +  2 

This  differential  equation  may  be  interpreted  by  noting  that  the  co- 
ordinates of  a  point  on  the  normal  at  (x,  y)  are 

f  =  x-Rzx/H,     7)  =  y-Rzy/H, 

where  R  is  the  distance  of  the  point  from  (x,  y).   If  now  two  consecutive 
normals  intersect  at  this  point,  we  have 

0  =  d£  =  dx  -  Ed  (zx/H)y     0  -  drj  =  dy  -  Rd  (zv/H), 
for  dR  =  0.    Expanding  in  the  form 

0  =  dx  [l  -  R  A  (z,///)]  -dyB  j-  (zv/H), 


and  eliminating  dx,  dy,  we  obtain  as  our  equation  for  R 

n       i       P  T^  (<>  fff\^.  ^  i 
0  =  l  -  R  [dx  (Z*IH)  +  dy  ( 

If  Rl  and  R2  are  the  roots,  we  have 


The  quantities  Rl  and  R2  are  called  the  principal  radii  of  curvature.    A 

minimal  surface  is  thus  characterised  by  the  equation  -^  +  p-  =  0  and 

/h     ^2 
a  surface  of  a  soap  film  subject  to  a  constant  pressure-difference  on  its  two 

sides  is  shaped  in  accordance  with  the  equation 

—-_[---=  constant. 

-«!        -#2 

When  the  film  is  subjected  to  only  a  smalLdifference  of  pressure  and  is 
stretched  across  a  hole  in  a  thin  flat  plate  we  can,  to  a  sufficient  approxi- 
mation, put  H  =  1  in  this  equation.  The  resulting  equation  is 


where  K  is  a  constant  and  the  boundary  condition  is  z  —  0  on  the  rim. 


Torsion  Problems  and  Soap  Films  171 

Now  the  same  differential  equation  and  boundary  conditions  occur  in 
a  number  of  physical  problems  and  a  soap-film  method  of  solving  such 
problems  in  engineering  practice  was  suggested  by  Prandtl  and  has  been 
much  developed  by  A.  A.  Griffith  and  G.  I.  Taylor*.  The  most  important 
problems  of  this  type  are : 

(1)  The  torsion  of  a  prism  ( Saint- Venant's  theory). 

(2)  The  flow  of  a  viscous  liquid  under  pressure  in  a  straight  pipe. 
These  problems  will  now  be  considered. 

EXAMPLES 

1.  The  forces  acting  on  the  rim  of  a  soap  film  of  tension  T  are  equivalent  to  a  force  F 
at  the  origin  and  a  couple  0.  Prove  that 


0=  f%T[rx  (nxds)}, 


where  the  vector  ds  denotes  a  directed  element  of  the  rim  and  the  vector  n  is  a  unit  vector 
along  the  normal  to  the  surface  of  the  film.  Show  by  transforming  these  integrals  into  surface 
integrals  that  the  force  and  couple  are  equivalent  to  a  system  of  normal  forces,  the  force 
normal  to  the  element  dS  being  of  magnitude 


2.  The  surface  of  a  film  closing  up  a  vessel  of  volume  V  can  be  regarded  as  one  of  a 
family  of  surfaces  for  which  C^  -f  C2  is  (7,  a  constant.  If  within  a  limited  region  of  space  there 
is  just  one  surface  of  this  family  that  can  be  associated  with  each  point  by  some  uniform  rule 
and  if  S'  is  another  surface  through  the  rim  of  the  hole,  e  the  angle  which  this  surface  makes 
at  a  point  (x,  y,  z)  with  the  surface  of  the  family  through  this  point,  the  area  of  the  outer 

surface  of  the  film  is  I  I  cos  c  .  dS'.   Hence  show  that  the  area  of  the  new  surface  is  greater 
than  that  of  the  film  if  it  encloses  the  same  volume. 

3.  If  w  =  zx,     t>  =  zv,    q***u*  +  ifi, 
show  that  the  variation  problem 

8  ffo(q)  dxdy  =  0 

leads  to  the  partial  differential  equation 


where  c2  [qQ"  (q)  -  Of  (q)]  =  q2  G'  (q). 

Show  also  that  the  two-dimensional  adiabatic  irrotational  flow  of  a  compressible  fluid 
leads  to  an  equation  of  this  type  for  the  velocity  potential  z,  the  function  G  (q)  being  given 
by  the  equation 

O  (q)  -  [2a2  +  (y  -  1)  (U*  -  fW~\ 
where  U,  a  and  y  are  constants. 

*  See  ch.  vn  of  the  Mechanical  Properties  of  Fluids  (Blackie  &  Son,  Ltd.,  1923). 


172    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

§  2-55.  The  torsion  of  a  prism.  Assuming  that  the  material  of  the  prism 
is  isotropic,  we  take  the  axis  of  z  in  the  direction  of  the  generators  of  the 
surface  and  consider  a  distortion  in  which  a  point  (#,  y,  z)  is  displaced  to 
a  new  position  (%  +  u,  y  +  v,  z  +  w),  where 

u  =  —  ryz,     v  =  rzx,     w  =  r</>, 

and  </>  is  a  function  of  x  and  y  to  be  determined.  The  constant  r  is  called 
the  twist.  This  distortion  is  supposed  to  be  produced  by  terminal  couples 
applied  in  a  suitable  manner  to  the  end  faces.  The  portion  of  the  surface 
generated  by  lines  parallel  to  the  axis  of  z  (the  mantle)  is  supposed  to  be 
free  from  stress.  These  are  the  simplifying  assumptions  of  Saint-Venant. 
It  is  easily  seen  that 

du  ___  dv  __  dw  _^dv      du  _ 
dx      dy      dz      dx     dy  ~~    ' 
du      dw          /dd> 

p      ____4_       __._,_       r 

**      dz^dx          (dx 
dw      dv 

p        —  _L  — 

vz      dy  ^  dz  ~ 

Hence,  if       Zx  =  fiezxy     Zy  =  ^eys,     Xx^  Yv  =  Zz  =  Xy  ==  0, 
the  equations  of  equilibrium 

az?  =  0    azy  =  0    az     az,  =  0 

dz          '       'dz          '       dx        dy 
show  that  Zx  and  Zy  are  independent  of  z  and  that 

9    /dJ>        \       d   /d(j>         \ 

>r  U    -  2/   +  3  -  U    +  #  )  =  0, 

dx  \dx      J  )      dy  \dy         J 

9¥     ,     9V          A 

d^+d^°- 

The  boundary  condition  of  no  stress  on  the  mantle  gives 

IZX  +  mZv  -  0, 

where  (I,  m,  0)  are  the  direction  cosines  of  the  normal  to  the  mantle  at 
the  point  (x,  y,  z). 

Let  us  now  introduce  the  function  </f  conjugate  to  </>,  then 
20  __  3i/r       d(f)  __      9i/f 
3x      dy  '      9?/  "~      9x  ' 


where  r2  =  a:2  +  y2.  The  boundary  condition  may  consequently  be  written 
in  the  form  ~  ~ 


Torsion  of  a  Prism  173 

where  %  =  iff  —  £r2  and  ds  is  a  linear  element  of  the  cross-section.  This 
equation  signifies  that  x  is  constant  over  the  boundary  and  so  the  problem 
may  be  solved  by  determining  a  potential  function  i/j  which  is  regular 
within  the  prism  and  which  takes  a  value  differing  by  a  constant  from  \r- 
on  the  mantle  of  the  prism.  Without  loss  of  generality  this  constant  may 
be  taken  to  be  zero  if  there  is  only  one  mantle. 

It  should  be  noticed  that  the  function  x  satisfies  the  equation 

32X  ,  92X__2 

3x2  T  dy2 

and,  with  the  above  choice  of  the  constant,  is  zero  on  the  mantle  when 
this  is  unique.  It  is  often  more  convenient  to  work  with  the  function  x> 
especially  as 


9v 

-  -" 

oy 
Since  x  vanishes  on  the  mantle  it  is  evident  that 

\\Zxdxdy  =  0,      \\Zydxdy  =  0. 

The  tractions  on  a  cross-section  are  thus  statically  equivalent  to  a 
couple  about  the  axis  of  z  of  moment 


M  =      (.rZy  -  yZ,)  dxdy  =  -  pr       *       +  y        dxdy. 

Integrating  by  parts  we  find  that 

M  = 

The  direction  of  the  tangential  traction  (Zx,  Zy)  across  the  normal 
section  of  the  prism  by  a  plane  z  =  constant  is  that  of  the  tangent  to  the 
curve  x  —  constant  which  passes  through  the  point.  The  curves  x  =  con- 
stant may  thus  be  called  "lines  of  shearing  stress."  The  magnitude  of  the 

traction  is  LLT  „-,  where  ~-  is  the  derivative  of  v  in  a  direction  normal  to 
r    3n  on  A 

the  line  of  shearing  stress. 

In  the  case  of  a  circular  prism 

x  =  i  (a2  -  r2), 
and*  in  the  case  of  an  elliptic  prism 


where  a  and  b  are  the  semi-axes  of  the  ellipse  and 


174    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

§  2-56.  Flow  of  a  viscous  liquid  along  a  straight  tube.  Consider  the 
motion  of  the  portion  contained  between  the  cross-sections  z  =  zl  and 
z  ==  z1  -f  h.  If  A  is  the  area  of  the  cross-section  and  />  the  density  of  the 
fluid,  the  equation  of  motion  is 

M*  ^  =  -4  (ft  -A)-  D, 

where  pl  and  p2  are  the  pressures  at  the  two  sections  and  D  is  the  total 
frictional  drag  at  the  curved  surface  of  the  tube.  If  u  is  the  velocity  of 
flow  in  the  direction  of  the  axis  of  z,  u  will  be  independent  of  z  if  the  fluid 
is  incompressible  and  so  we  may  write 

u  =  u(x,  y,  t). 

We  now  introduce  the  hypothesis  that  there  is  a  constant  coefficient 
of  viscosity  p,  such  that 


where  ~—  denotes  a  differentiation  in  the  direction  of  the  normal  to  the 
on 

surface  of  the  tube.  Transforming  the  surface  integral  into  a  volume 
integral,  we  have  the  equation  of  motion 

A,du      Al  v  ,    ffr    /92^     3^\  7    7 

PAh  -ft  =  A  (ft  -  ft)  +  J  J  V  ^2  +  gp)  dxdy. 

Since  h  is  arbitrary  this  may  be  written  in  the  form 
Su          dp 

- 


When  the  motion  is  steady  this  equation  takes  the  form 


where  —  2Jf£  =  -  >p  and  can  be  regarded  as  a  constant,  because  u  is  in- 

dependent of  z.   This  is  the  equation  used  by  Stokes  and  Boussinesq. 
In  the  case  of  an  elliptic  tube 


where 

For  an  annular  tube  bounded  by  the  cylinders  r  =  a,  r  =  b  we  may 

tt  =  JA:  (a2  -  r2)  +  JJP  (62  -  a2)  log  (6/r). 
The  total  flux  $  is  in  this  case 

n      _    f6       ,       wJfa«f  .      (s2-  I)2)          ,       ,.  . 

«  -  2w  Ja  urdr  -  -4-  r  ~  log  .  r    (5  =  6/a) 


Rectilinear  Viscous  Flow  175 

and  so  the  average  velocity  is 


4     - 

If  there  is  no  pressure  gradient  the  equation  of  variable  flow  is 

du 


where  v  =  /x/p.  This  equation  is  the  same  as  the  equation  of  the  conduction 
of  heat  in  two  dimensions.  The  fluid  may  be  supposed  in  particular  to  lie 
above  a  plane  z  =  0  which  has  a  prescribed  motion,  or  to  lie  between  two 
parallel  planes  with  prescribed  motions  parallel  to  their  surfaces. 

The  simple  type  of  steady  motion  of  a  viscous  fluid  which  is  given  by 
the  equation  (I)  does  not  always  occur  in  practice.  The  experiments  of 
Osborne  Reynolds,  Stanton  and  others  have  shown  that  when  a  viscous 
fluid  flows  through  a  straight  pipe  of  circular  section  there  is  a  certain 
critical  velocity  (which  is  not  very  definite)  above  which  the  flow  becomes 
irregular  or  turbulent  and  is  in  no  sense  steady.  From  dimensional  reason- 
ing it  has  been  found  advantageous  to  replace  the  idea  of  a  critical  velocity 
by  that  of  a  critical  dimensionless  quantity  or  Reynolds  number  formed 
from  a  velocity,  a  length  and  the  kinematic  viscosity  v  of  the  fluid.  In  the 
case  of  flow  through  a  pipe  the  velocity  V  may  be  taken  to  be  the  mean 
velocity  over  the  cross-section,  the  length,  the  diameter  of  the  pipe  (d). 
For  steady  "laminar  flow"  the  ratio  Vd/v  must  not  exceed  about  2300. 

In  the  case  of  the  motion  of  air  past  a  sphere  a  similar  Reynolds 
number  may  be  defined  in  which  d  is  the  diameter  of  the  sphere.  In  order 
that  the  drag  may  be  proportional  to  the  velocity  V  the  ratio  Vd/v  must 
be  very  small. 

EXAMPLES 

1.   In  viscous  flow  between  parallel  planes  x  =  ±  a  the  velocity  is  given  by  an  equation 


where  c  is  the  maximum  velocity.    Prove  that  the  mean  velocity  is  two-thirds  of  the 
maximum. 

2.  In  a  screw  velocity  pump  the  motion  of  the  fluid  is  roughly  comparable  with  that  of 
a  viscous  liquid  between  two  parallel  planes  one  of  which  moves  parallel  to  the  other  and 
drags  the  fluid  along,  although  there  is  a  pressure  gradient  resisting  the  flow.   Calculate 
the  efficiency  of  the  pump  and  find  when  it  .is  greatest. 

Work  out  the  distribution  of  velocity  and  the  efficiency  when  the  machine  acts  as  a 
motor,  that  is,  when  the  fluid  is  driven  by  the  pressure  and  causes  the  motion  of  the  upper 
plate. 

[Rowell  and  Finlayson,  Engineering,  vol.  cxxvi.  p.  249  (1928).] 

3.  The  Eulerian  equation  associated  with  the  variation  problem 


18 


176    Applicatidns  of  the  Integral  Theorems  of  Gauss  and  Stokes 

L.  Lichtenstein  [Math.  Ann.  vol.  LXIX,  p.  514,  1910]  has  shown  that  when  f(x,  y)  is  merely 
continuous  there  may  be  a  function  u  which  makes  8/  =  0  and  does  not  satisfy  the  Eulerian 

equation. 

§  2*57.  The  vibration  of  a  membrane.  Let  T  be  the  tension  of  the 
membrane  in  the  state  of  equilibrium  and  w  the  small  lateral  displacement 
of  a  point  of  the  membrane  from  the  plane  in  which  the  membrane  is 
situated  when  in  a  state  of  equilibrium,  the  vibrations  which  will  be  con- 
sidered are  supposed  to  be  so  small  that  any  change  in  area  produced  by 
the  deflections  w  does  not  produce  any  appreciable  percentage  variation 
of  T.  The  quantity  T  is  thus  treated  as  constant  and  the  potential  energy 

,  (Sw\*  ,   /3w\2l*  ,    , 
+  (  *-)   +  (  *  dxdy 

\dxj        \dyj  J  y 

is  replaced  by  the  approximate  expression 


Let  pdxdy  be  the  mass  of  the  element  dxdy.  The  equation  of  motion 
of  the  membrane  will  be  obtained  by  considering  the  variation  of 


where 

„      Iff    /dw\2  ,    j        r7 
IS  =  -  \\  pi  -    I  dxdy,     V  - 

The  integral  to  be  varied  is  thus 

=  2jJJ  L^  \dt) 

where  w  -=  0  on  the  boundary  curve  for  all  values  of  t.  The  Eulerian 
equation  of  the  Calculus  of  Variations  gives 


where  c2  =  T/p. 

This  is  the  equation  of  a  vibrating  membrane.  The  equation  occurs  also 
in  electromagnetic  theory  and  in  the  theory  of  sound.    In  the  case  when 

w  is  of  the  fotm  .  .     ^ 

w  ==  sin  Ky  .  v  (x,  t), 

the  function  v  satisfies  the  equation 

^v        . 

ri  ~  K  v 

1x2 

which  is  of  the  same  form  as  the  equation  of  telegraphy. 

It  should  be  noticed  that  a  corresponding  variation  principle 


m  m  ,    ,    ,    , 

o-     -T\i~)  -yhr)  \dxdydzdt=*Q 

SxJ  \dyj  \3zjj        y 


The  Equation  of  Vibrations  111 

gives  rise  to  the  familiar  wave-equation 

d2w 


which  governs  the  propagation  of  sound  in  a  uniform  medium  and  the 
propagation  of  electromagnetic  waves.  A  function  w  which  satisfies  this 
equation  is  called  a  wave-function. 

Love  has  shown*  that  the  equation  (A)  occurs  in  the  theory  of  the 
propagation  of  a  simple  type  of  elastic  wave. 

Taking  the  positive  direction  of  the  axis  of  z  upwards  and  the  axis  of 
x  in  the  direction  of  propagation,  we  assume  that  the  transverse  displace- 
ment v  is  given  by  the  equation 

v  =  Y  (z)  cos  (pt  —  fx). 

The  components  of  stress  across  an  area  perpendicular  to  the  axis  of 
y  are  . 

V  °v       Y    -  n      v  —     - 

x^^dx'    -z*~u>      z~^dz 

respectively  and  so  the  equation  of  motion 


*v     dY,     dY,  ,     dY 


. 


(B) 


p'dt*  " 
takes  the  form 


When  p  and  JJL  are  constants  this  is  the  same  as  the  equation  of  a 
vibrating  membrane,  but  when  p  and  //.  are  functions  of  z  the  equation  is 
of  a  type  which  has  been  considered  by  Meissnerf. 

Transverse  waves  of  this  type  have  been  called  by  JeffreysJ  "Love 
waves,"  they  are  of  some  interest  in  connection  with  the  interpretation  of 
the  surface  waves  which  are  observed  after  an  earthquake. 

It  may  be  mentioned  that  the  general  equation  (B)  may  be  obtained 
by  considering  the  variation  of  the  integral 

,      Iff  IT    iSv 

J-2lJJKai 

and  an  extension  can  be  made  to  the  case  in  which  p  and  JJL  are  functions 
of  x,  z  and  t. 

§  2-58.  The  electromagnetic  equations.  Consider  the  variation  of  the 
integral 

n\\  Ldxdydzdt, 

*  Some  Problems  of  Qeodynamics,  p.  160  (Cambridge  University  Press,  1911). 

t  Proceedings  of  the  Second  International  Congress  for  Applied  Mathematics  {Zurich,  1926). 

{  The  Earth,  p.  165  (Cambridge  University  Press,  1924). 


178    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 
where  2L  =  Hx*  +  Hv*  +  Ht*  -  E*  -  Ev*  -  E*, 


and 

H 


V 


-fc--fa,        ^---g--^,  (A) 

MV  __  ?4*       F  =  -  ~A* 
"  dx       dy  9       *  dt 

If  the  variations  of  Ax,  AV)  Az  and  0  vanish  at  the  boundary  of  the 
region  of  integration,  the  Eulerian  equations  give 

~dy"~"dz  ^"dT9 

~dz~~  8^r=="aT5 

~dx  ~  ~dy^~df9 

dx        dy        dz 
In  vector  notation  these  equations  may  be  written 

curlH=-~,     divE=0,  (B) 

and  equations  (A)  take  the  form 

H=curlA,    E«-~-V*.  (C) 

ot 

These  equations  imply  that 

!=-??,    divH=0.  (D) 


The  two  sets  of  equations  (B)  and  (D)  are  the  well-known  equations  of 
Maxwell  for  the  propagation  of  electromagnetic  waves  in  the  ether;  the 
unit  of  time  has,  however,  been  chosen  so  that  the  velocity  of  light  is 
represented  by  unity.  The  foregoing  analysis  is  due  essentially  to  Larmor. 

Writing  Q  =  H  +  iE,  the  two  sets  of  equations  may  be  combined 
into  the  single  set  of  equations 

divQ=0.  ......  (E) 


=-, 
ot 

By  analogy  with  (C)  we  may  seek  a  solution  for  which 

Q=-icurlL=-~?~VA.  ......  (F) 

The  relations  between  L  and  A  may  be  satisfied  by  writing 


,  --,        ......  (G) 


Electromagnetic  Equations  179 

where  G  is  a  complex  vector  of  type  T  +  tH,  while  T  and  II  are  real  '  Hertzian 
vectors  '  whose  components  all  satisfy  the  wave-equation 


When  we  differentiate  to  find  an  expression  for  Q  in  terms  of  G  and  K 
the  terms  involving  K  cancel  and  we  find  the  Righi-Whittaker  formulae 


H  =  curl(^curir-f  ^),   \  (I) 

E  =  curl  (curl  II  -  g 
If  L  =  B  +  iA,  A  =  Y  -f-  iO,  where  A,  B,  O  and  XF  are  real,  we  have 


......  (J) 

E=-curlB=-8£-  V<D, 

VI 

curir,     B-- 


O  =  -  div  n,     T  =  -  div  T,       J 

where  A,  B,  O  and  Y  are  wave-functions  which  are  connected  by  the 
identical  relations  ^  ^^ 

=  0.  ......  (L) 


The  corresponding  formulae  for  the  case  in  which  the  unit  of  time  is 
not  chosen  so  that  the  velocity  of  light  is  unity  are  obtained  from  the 
foregoing  by  writing  ct  in  place  of  t  wherever  t  occurs. 

If  we  write  Q'  =  eieQ,  where  8  is  a  constant,  it  is  evident  that  the  vector 
Q'  satisfies  the  same  differential  equations  as  Q  and  can  therefore  be  used 
to  specify  an  electromagnetic  field  (Ex,  Hx)  associated  with  the  original 
field  (E,  H).  It  will  be  noticed  that  the  function  L'  for  this  associated  field 
is  not  the  same  as  L,  for 

L9  =  H'2  -  E"*  =  (H2  -  E2)  cos  20-2(E.H)  sin  20. 

Also  (W  .  H')  =  (H2  -  E*)  sin  20  +  2  (E  .  H  )  cos  26. 

There  are,  however,  certain  quantities  which  are  the  same  for  the  two 
fields.  These  quantities  may  be  defined  as  follows  : 


o         JP    u        i?  17        n 
tox  =  &yJlz  —  &ZHV  =  Ux, 

XX  =  EX*  +  HX*-W,         \  (M) 

Y,  =  EVE,  +  HVHZ  =  ZV,\ 


180    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

It  is  interesting  to  note  that  these  quantities  may  be  arranged  so  as  to 
form  an  orthogonal  matrix  * 


X 


iSx    iSv 
We  have,  in  fact,  the  relations 


x, 

Y, 
Z2 

iS. 


w 


where 


T/f/2  C*    2  Q    2  C  2  T 

rr      —  &x     —   &y     —  &z     ~  •*•  i 

Xx  Yx  +  AV  Y y  -f*  -Xz  •*  3  —  GxGy  =  0, 

V     O        i       V     O        i       V    O       i     /"Y     TI/          A 
Aa-Oa-  +  AVOV  +  A20Z  +   C^a;  W    =   U, 

I=l(H*-  E*)*+  (E.Hy2- 


.(N) 


§  2-59.  TAe  conservation  of  energy  and  momentum  in  an  electromagnetic 
field.  It  follows  from  the  field  equations  (B)  and  (D)  that  the  sixteen  com- 
ponents of  the  orthogonal  matrix  satisfy  the  equations 

PY      ajr      8X,     8ga  _ 

a^     'dt  ~~  J 


dx^  dy 


dx    ' 
"^      i 


57_v  ayz 

3y  3z 

rjiy  ^^ 

C£jy  Ofj^ 

dy  dz 


dy 


dt 

d_G_> 
"dt 

dw 


~dz 


=  0 


=  0. 


.(A) 


Regarding  Sx,  Sy,  Sz  for  the  moment  as  the  components  of  a  vector  S 
and  using  the  suffix  n  to  denote  the  component  along  the  outward-drawn 
normal  to  a  surface  element  da  of  a  surface  or,  we  have 


div/S.^T 
dW 


(dr  —  dxdydz) 


dt 


dr 


=  -  £          ^rfr. 


In  this  equation  the  region  of  integration  is  supposed  to  be  such  that  the 
derivatives  of  8  and  W  in  which  we  are  interested  are  continuous  functions 
of  x,  y,  z  and  t.  This  will  certainly  be  the  case  if  the  field  vectors  and  their 
first  derivatives  are  continuous  functions  of  x,  y,  z  and  t. 


*  H.  Minkowski,  Gott.  Nachr.  (1908). 


Conservation  of  Energy  and  Momentum  181 

Let  us  now  regard  W  as  the  density  of  electromagnetic  energy  and  S 
as  a  vector  specifying  the  flow  of  energy,  then  the  foregoing  equation  can 
be  interpreted  to  mean  that  the  energy  gained  or  lost  by  the  region  en- 
closed by  a  is  entirely  accounted  for  by  the  flow  of  energy  across  the 
boundary.  This  is  simply  a  statement  of  the  Principle  of  the  Conservation 
of  Energy  for  the  electromagnetic  energy  in  the  ether. 

The  equations  involving  Ox ,  Gv ,  Gz  may  be  regarded  as  expressing  the 
Principle  of  the  Conservation  of  Momentum.  We  shall,  in  fact,  regard  Gx 
as  the  density  of  the  ^-component  of  electromagnetic  momentum  and 
(—  Xx,  —  XV)  —  Xz)  as  the  components  of  a  vector  specifying  the  flow  of 
the  ^-component  of  electromagnetic  momentum. 

The  vector  S  is  generally  called  Poynting's  vector  as  it  was  used  to 
describe  the  energy  changes  by  J.  H.  Poynting  in  1884.  The  vector  G  was 
introduced  into  electromagnetic  theory  by  Abraham  and  Poincare. 

In  the  case  of  an  electrostatic  field 


if  there  are  only  volume  charges  and  the  first  integral  is  taken  over  all 
space,  for  then  the  surface  integral  may  be  taken  over  a  sphere  .of  infinite 
radius  and  may  be  supposed  to  vanish  when  the  total  amount  of  electricity 
is  finite  and  there  is  no  electricity  at  infinity.  It  should  be  noticed  that  in 
the  present  system  of  units  Poisson's  equation  takes  the  form 

V2</>  -f  p  =  0, 

where  p  is  the  density  of  electricity.  When  there  are  charged  surfaces  an 
integral  of  type 


must  be  added  to  the  right-hand  side  for  each  charged  surface. 

The  new  expression  for  the  total  energy  may  be  written  in  the  form 

U  = 


This  may  be  derived  from  first  principles  if  it  is  assumed  that  <f)8e  is  the 
work  done  in  bringing  up  a  small  charge  8e  from  an  infinite  distance 
without  disturbing  other  charges.  Now  suppose  that  each  charge  in  an 
electrostatic  field  is  built  up  gradually  in  this  way  and  that  when  an 
inventory  is  taken  at  any  time  each  carrier  of  charge  has  a  charge  equal 
to  A  times  the  final  amount  and  a  potential  equal  to  A  times  the  final 


182    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

potential.  A  being  the  same  for  all  carriers.  As  A  increases  from  A  to  A  -f  dX 
the  work  done  on  the  system  is 

dU  =  2  (Ac/>)  ed\. 
Integrating  with  respect  to  A  between  0  and  1  we  get 

U  =  iSe</>. 

The  carriers  mentioned  in  the  proof  may  be  conducting  surfaces  capable 
(within  limits)  of  holding  any  amount  of  electricity.  If  the  carriers  are 
taken  to  be  atoms  or  molecules  there  is  the  difficulty  that,  according  to 
experimental  evidence,  the  charge  associated  with  a  carrier  can  only  change 
by  integral  multiples  of  a  certain  elementary  charge  e.  For  this  reason  it 
seems  preferable  to  start  with  the  assumption  that  W  represents  the  density 
of  electromagnetic  energy. 

On  account  of  the  symmetrical  relations 

72  =  Zv,etc.,     Sx=  O,,  etc., 
we  can  supplement  the  relations  (A)  by  six  additional  equations  of  types 


.  (yZx  -zYx)  +      (yZv  -zYv)  +      (yZz  -zY,)-      (yG,  -  zGv)  =  0, 


a    (x8x  +  tX.)  +       (xSv  +  tXv)  +  -    (x8.  +  tX.)  +     (xW  -  tO.)  =  0. 

The  equations  of  the  first  type  may  be  supposed  to  express  the  Principle 
of  the  Conservation  of  Angular  Momentum.  We  shall,  in  fact,  regard 
yGz  —  zOy  as  the  density  of  the  ^-component  of  angular  momentum  and 
(zYx  —  yZx,  zYv  —  yZy,  zYz  —  yZz)  as  the  components  of  a  vector  which 
specifies  the  flow  of  the  ^-component  of  angular  momentum. 

The  equations  of  the  second  type  are  not  so  easily  interpreted.  We  shall, 
however,  regard  xW  —  tOx  as  the  density  of  the  moment  of  electromagnetic 
energy  with  respect  to  the  plane  x  =  0.  This  quantity  is,  in  fact,  analogous 
to  Smo;,  a  quantity  which  occurs  in  the  definition  of  the  centre  of  mass  of  a 
system  of  particles.  Here  and  in  the  relation  S  —  G  we  have  an  indication 
of  Einstein's  relation 

(Energy)  =  (Mass)  (square  of  the  velocity  of  light) 
which  is  of  such  importance  in  the  theory  of  relativity. 

The  quantities  (xSx  -f  tXX)  xSv  -f  tXv,  xSz  -f  tX9)  will  be  regarded  as 
the  components  of  a  vector  which  specifies  the  flow  of  the  moment  of 
electromagnetic  energy  with  respect  to  the  plane  x  =  0.  The  equation  may, 
then,  be  interpreted  to  mean  that  there  is  conservation  of  the  moment 
with  respect  to  the  plane  x  =  0.  There  is,  in  fact,  a  striking  analogy  with 
the  well-known  principle  that  the  centre  of  mass  of  an  isolated  mechanical 
system  remains  fixed  or  moves  uniformly  along  a  straight  line*. 

*  A.  Einstein,  Ann.  d.  Physik  (4),  Bd.  xx,  S.  627  (1906);  G.  Herglotz,  ibid.  Bd.  xxxvi,  S.  493 
(1911);  E.  Bessel  Hagen,  Math.  Ann.  Bd.  ijcxxiv,  S.  268  (1921). 


Conservation  of  Angular  Momentum  183 

EXAMPLES 

1.  Prove  that  when  there  are  no  external  forces  the  equations  of  motion  of  an  incom- 
pressible inviscid  fluid  of  uniform  density  give  the  following  equations  which  express  the 
principles  of  the  conservation  of  momentum  and  angular  momentum,  the  motion  being  two- 
dimensional  : 


-  uy)  -  yp]  +  g-  [pv(vx  -  uy)  +  xp]  -  0. 

Hence  show  that  the  following  integrals  vanish  when  the  contour  of  integration  does  not 
contain  any  singularities  of  the  flow  or  any  body  which  limits  the  flow,  the  motion  being 
steady: 

I  p  (v  -f  iu)  (id  +  vm)  ds  +  I  p  (m  +  il)  ds, 


I  p  (xv  —  yu)  (ul  +  vm)  ds  -f  I  p  (xm  —  yl)  ds. 


When  the  contour  does  contain  a  body  limiting  the  flow  the  integrals  round  the  contour 
are  equal  to  corresponding  integrals  round  the  contour  of  the  body. 

2.   Let  u  (#! ,  x2 ,  ...  xn)  be  a  function  which  is  to  be  determined  by  a  variational  principle 
8/  =r  0,  where 

r  r       r  * 

I  =  II  ...  M(xl9x29  ...xnfu9ultu29  ...un)dxlidx^...dxn 

and  ur  =  -  - .    Suppose  further  that  7  is  unaltered  in  value  by  the  continuous  group  of 
transformations  whose  infinitesimal  transformation  is 


(r  =  1,  2,  ... 


_  n 

and  let  Bu  =  Aw  —    2   wrAxr, 

r-l 


—         n    dR 

then  08u-    2   ^T, 

r-i  a*r 

where  5r  -  -  f&xr  -  ^  8u        (r  «  1,  2,  ...  w). 

When  the  function  w  satisfies  the  Eulerian  equation  0  =  0  the  foregoing  result  gives  a  set  of 
equations  of  conservation.  [E.  Noether,  Gott.  Nachr.  p.  238  (1918).] 

3.  If  n  =  2,  /  =  (u^  —  <w22  -h  )3w2)  ey*2  where  o#  )5  and  y  are  arbitrary  constants,  we  may 
write  Aa^  =  ea,  A#2  =  <2>  ^w  —  —  iv€2tt  w^re  ex  and  ea  are  two  independent  small  quantities 
whose  squares  and  products  may  be  neglected.  Hence  show  that  the  differential  equation 
uu  =  aw22  +  yaw2  4-  /to  leads  to  two  equations  of  conservation 

A  {2^1^  +  V^}  -  (A  +  y)  K2  +  aW22  +  ^w2  -f 
J5j  {V  +  aw22  -  /3w2}  =  (D2  -f 


where  Z^EE  /-,  D2=  =—  .       [E.  T.  Copson,  Proc.  ^rfm.  Jfefa^.  Soc.  vol.  XLII,  p.  61  (1924).] 
dx^  0X2 


184     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 
§  2-61.    Kirchhoff'  s  formula.  This  theorem  relates  to  the  equation 

D2"  +  a  (x,  y,  z,  t)  =  0,  ......  (A) 

and  to  integrals  of  type 

u  =--  I  r~lf  (t  -  r/c)  F  (XQ,  y0,  z0)  dr0. 

Let  us  suppose  that  throughout  a  specified  region  of  space  and  a  specified 
interval  of  time,  u  and  its  differential  coefficients  of  the  first  order  are 
continuous  functions  of  x,  y,  z  and  t  ;  let  us  suppose  also  that  the  differential 

d2u    d2u 
coefficients  of  the  second  order  such  as  ~  „  ;  ,  ~  2  and  the  quantity  a  are 

finite  and  integrable. 

Let  Q  be  any  point  (x0,  ?/0,  z0)  which  need  not  be  in  the  specified  region 
of  space  and  consider  the  function  v  derived  from  u  by  substituting  t  —  r/c 
in  place  of  t,  r  denoting  the  distance  from  Q  of  any  point  (x,  y,  z)  in  the 
specified  region.  It  is  easy  to  verify  that  v  satisfies  the  partial  differential 

equation 

__0        2r  f  9  (x  dv\        d  /y  Sv\       3  /z 


where  [a]  denotes  the  function  derived  from  a  by  substituting  t  —  r/c  in 
place  of  t. 

We  now  multiply  the  above  equation  by     and  integrate  it  throughout 

a  volume  lying  entirely  within  the  specified  region  of  space.  The  volume 
integral  can  then  be  split  into  two  parts,  one  of  which  can  be  transformed 
immediately  into  an  integral  taken  over  the  boundary  of  this  region.  Let 
the  point  Q  be  outside  the  region  of  integration,  then  we  have 

3  /1\        I3v        2dr  dv]    ,0       f  [a]  , 
V  ^  }-  -   *  ---  cf    ^  \  dS  +       —  dr' 

[_    dn\rj       rdn      crdndt]  }    r 

When  Q  lies  within  the  region  of  integration  the  volume  may  be  sup- 
posed to  be  bounded  externally  by  a  closed  siirface  S1  and  internally  by  a 
small  closed  surface  S2  surrounding  the  point  Q.  Passing  to  the  limit  by 
contracting  82  indefinitely  the  value  of  the  integral  taken  over  S2  is 
eventually 


ff  f 
-    \\\ 

JJ  [_ 


wjjiere  VQ  denotes  the  value  of  v  at  Q  and  this  is  the  same  as  the  value  of 
u  at  Q.   Hence  in  this  case 

fW^      flT  a  fl\     ldv     2dvdr']jv 

±TtUQ  ==  rfr  -  Vx  -  (  -  )  --  a  -  -      -^5~\dS. 

Q      I    r  JJ  l  dn\rj      rdn      crdtdn] 

dv      [du~]      dr  [dul 
Now 


Kirchhoff's  Formula  185 

hence  finally  we  have  Kirchhoff's  formula* 

f  M  7         ff  (r   n  8  /1\       1  [dul        1  dr  rSi^l)    7r> 

4rrwQ  =     LJdr  -     N[w]  =-  (-    -      U-    --  U,       dS, 

Q      1    r  jj  \L   *dn\r)      r  [dn]      crdn[dt]\ 

where  a  square  bracket  [/]  indicates  that  the  quantity /is  to  be  calculated 
at  time  t  —  r/c.  When  the  point  Q  lies  outside  the  region  of  integration 
the  value  of  the  integral  is  zero  instead  of  UQ  . 

When  u  and  a  are  independent  of  t  the  formula  becomes 

47TUQ  =    l-dr  —    \\   \U~-  (-} =-  ,»  , 

J  r  ]J(dn  \rj      rdn) 

and  the  equation  for  u  is 

V*u  +  a  (x,  y,  z)  =  0. 

If  we  make  the  surface  Sl  recede  to  infinity  on  all  sides  the  surface 
integrals  can  in  many  cases  be  made  to  vanish.  We  may  suppose,  for 
instance,  that  in  distant  regions  of  space  the  function  u  has  been  zero  until 
some  definite  instant  tQ.  The  time  t  —  r/c  then  always  falls  below  tQ  when 
r  is  sufficiently  large  and  so  all  the  quantities  in  square  brackets  vanish. 

rs 

The  surface  integral  also  vanishes  when  u  and  ~    become  zero  at  infinity 

and  tend  to  zero  as  r  ->  oo  in  such  a  way  that  u  is  of  order  r~l  and  ~- ,  ~- 
of  order  r~2.  In  such  cases  we  have  the  formula 

477^—  dr,  (B) 

where  the  integral  is  extended  over  all  the  regions  in  which  the  integrand 
is  different  from  zero. 

If  [a]  exists  only  within  a  number  of  finite  regions  which  do  not  extend 
to  infinity  the  function  UQ  defined  by  this  integral  possesses  the  property 
that  UQ  ->  0  like  r0~l  as  r0  ->  oo,  r0  being  the  distance  from  the  origin  of 

co-ordinates,  but  it  is  not  always  true  that  -~  is  of  order  r0~2.  To  satisfy 

this  condition  we  may,  however,  suppose  that  ^-  is  zero  for  values  of  t 

*"*>  — i 
less  than  some  value  t0 .  Then  if  r  is  sufficiently  large    ^      is  zero  because 

—  r/c  falls  below  t0 . 

Wave-potentials  of  type  (B)  are  called  retarded  potentials;  the  analysis 
shows  that  they  satisfy  the  equation  (A)  and  that  the  surface  integral 

ffkilfiU-1^!-1*:^!! 

JJ  (     Jdn\rJ     r  [dn]      crdn[dt]\ 


*  G.Kirchhoff,  Berlin.  Sitzungsber.  S.  641  (1882);  Wied.  Ann.  Bd.  xvni (1883);  Qes.  AM.  Bd.  u, 
S.  22.  The  proof  given  in  the  text  is  due  substantially  to  Beltrami,  Rend.  Lined  (5),  t.  iv  (1895), 
and  is  given  in  a  paper  by  A.  E.  H.  Love,  Proc.  London  Math.  Soc.  (2),  vol.  i,  p.  37  (1^03).  An 
extension  of  Kirchhoff's  formula  which  is  applicable  to  a  moving  surface  has  been  given  recently 
by  W.  R.  Morgans,  Phil.  Mag.  (7),  vol.  ix,  p.  141  (1930). 


186    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

represents  a  solution  of  the  wave-equation  except  for  points  on  the  surface 
S,  for  this  integral  survives  when  we  put  a  =  0.    It  should  be  noticed, 

however,  that  when  we  put  a  =  0  the  quantities  [u]  ,    «-    ,     ^-    become 

those  relating  to  a  wave-function  u  which  is  supposed  in  our  analysis  to 
exist  and  to  satisfy  the  postulated  conditions.    When  the  quantities  [u]> 

21  >  12"    are  c"losen  arbitrarily  but  in  such  a  way  that  the  surface  integral 

exists  it  is  not  clear  from  the  foregoing  analysis  that  the  surface  integral 
represents  a  solution  of  the  wave-equation.    If,  however,  the  quantities 

[u]  ,  Ur-  L     ^r    possess  continuous  second  derivatives  with  respect  to  the 

time  the  integrand  is  a  solution  of  the  wave-equation  for  each  point  on 
the  surface.   It  can,  in  fact,  be  written  in  the  form 


where  [u]  -/*-,  =  ?   '  ~      '    Now       stands  f  or 


7a        3       3 

ZQ-+    W    =-    +    7&=-    , 

d#  <7J/  C7Z 

where  I,  m  and  ?i  are  constants  as  far  as  x,  y  and  z  are  concerned  and  each 
term  such  as  ^-    -/(£  --  )    is  a  solution  of  the  wave-equation;  consequently 

OX  \Jf      \          C/  J 

the  whole  integrand  is  a  solution  of  the  wave-equation  and  it  follows  that 
the  surface  integral  itself  is  a  solution  of  the  wave-equation. 

In  the  special  case  when  a  and  u  are  independent  of  t  we  have  the  result 
that  when  or  satisfies  conditions  sufficient  to  ensure  the  existence  and 
finiteness  of  the  second  derivatives  of  V  (see  §  2'  32)  the  integral 


r 
is  a  solution  of  Poisson's  equation 

V2F  -f  47T(T  (x,  y,  z)  =  0, 

and  the  integral  U  =  f  [  \u  J-  (-}  -  -  ^1  dS 

e  JJ  {    dn\rj      rdn) 

is  a  solution  of  Laplace's  equation. 

§  2-62.  Poisson's  formula.  When^he  surface  Sl  is  a  sphere  of  radius  ct 
with  its  centre  at  the  point  Q,  [u]  denotes  the  value  of  u  at  time  t  =  0  and 
Kirchhoff  's  formula  reduces  to  Poisson's  formula  * 


*  The  details  of  the  transformation  are  given  by  A.  E.  H.  Love,  Proc.  London  Math.  Soc.  (2), 
vol.  I,  p.  37  (1903). 


Poisson's  Formula  187 

where  /,  g  denote  the  mean  values  of  /,  g  respectively  over  the  surface  of 
a  sphere  of  radius  ct  having  the  point  (xy  y,  z)  as  centre  and  u  is  a  wave- 
function  which  satisfies  the  initial  conditions 

u=f(x,y9z),     ^  =0(z,2/,z), 
when  t  =  0. 

If  we  make  use  of  the  fact  that  each  of  the  double  integrals  in  Poisson's 
formula  is  an  even  function  of  t  we  may  obtain  the  relation* 


^  u(x,y,z,t)dt=f. 
This  relation  may  be  written  in  the  more  general  form 


^J      u(x,y,z,8)ds 

1      ffff2ir 

==  j-          u  (x  +  CT  sin  6  cos  <f>,  y  -f  CT  sin  0  sin  <f>,  z  -f  CT  cos  0,  £)  sin  9ddd<f>. 

47T  J  0  J  0 

When  w  is  independent  of  the  time  this  equation  reduces  to  Gauss's  well- 
known  theorem  relating  to  the  mean  value  of  a  potential  function  over  a 
spherical  surface. 

If  u  (x,  y,  z}  s)  is  a  periodic  function  of  s  of  period  2r,  where  T  is  in- 
dependent of  x,  y  and  z,  the  function  on  the  left-hand  side  is  a  solution  of 
Laplace's  equation,  for  if 


rt  +  r 
V  =  c2        u(x,  y,  z,  s)  ds, 

Jt-T 


we  have  V2F  =  c2         Vhids  =         ^  9  d«9  =  0. 


It  then  follows  that  the  double  integral  on  the  right-hand  side  is  also  a 
solution  of  Laplace's  equation. 

If  in  Poisson's  formula  the  functions  /  and  g  are  independent  of  z  the 
formula  reduces  to  Parseval's  formula  for  a  cylindrical  wave-function. 
Since  we  may  write 

c2*2  sin  Oddd<f>  =  da  .  sec  6  =  ct  (cH2  -  />2)~*  da, 

where  da  is  an  element  of  area  in  the  #y-plane  and  p  the  distance  of  the 
centre  of  this  element  from  the  projection  of  the  centre  of  the  sphere,  we 
find  that 

277  .  u  (x,  y,  *)  =  I  JJ  da  .  (cH*  -  p2)^/  (x  4-  f  ,  y  4-  ij) 

da  .  (cW  -  p2)'*  g  (x  +  £,  y  +  r?), 


where  da  =  d^d-q  and  the  integration  extends  over  the  interior  of  the  circle 

2  2 


Cf.  Rayleigh's  Sound,  Appendix. 


188    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

This  formula  indicates  that  the  propagation  of  cylindrical  waves  as 
specified  by  the  equation  G2w  =  0  is  essentially  different  in  character  from. 
that  of  the  corresponding  spherical  waves.  In  the  three-dimensional  case 

the  value  of  a  wave-function  u  (x,  y,  z,  t)  at  a  point  (#,  y,  z)  at  time  t  is 

fjfj 
completely  determined  by  the  values  of  u  and  ^-  over  a  concentric  sphere 

ot 

of  radius  cr  at  time  t  —  T.  If  a  disturbance  is  initially  localised  within  a 
sphere  of  radius  a  then  at  time  t  the  only  points  at  which  there  is  any 
disturbance  are  those  situated  between  two  concentric  spheres  of  radii 
ct  -f-  a  and  ct  —  a  respectively,  for  it  is  only  in  the  case  of  such  points  that 
the  sphere  of  radius  ct  with  the  point  as  centre  will  have  a  portion  of  its 
surface  within  the  sphere  of  radius  a.  This  means  that  the  disturbance 
spreads  out  as  if  it  were  propagated  by  means  of  spherical  waves  travelling 
with  velocity  c  and  leaving  no  residual  disturbance  as  they  travel  along. 
In  the  two-dimensional  case,  on  the  other  hand,  the  value  of  u  (x,  y,  t) 

at  a  point  (x,  y)  at  time  t  is  not  determined  by  the  values  of  u  and  -^-  over 

a  concentric  circle  of  radius  CT  at  time  t  —  T.  To  find  u  (x,  y,  t)  we  must 

f)tj 
know  the  values  of  u  and  -^-  over  a  series  of  such  circles  in  Which  r  varies 

ot 

from  zero  to  some  other  value  rl  .  If  the  initial  disturbance  at  time  t  =  0  is 
located  within  a  circle  of  radius  a,  all  that  we  can  say  is  that  the  disturbance 
at  time  t  is  located  within  a  circle  of  radius  ct  +  a  and  not  simply  within 
the  region  between  two  concentric  circles  of  radii  ct  -j-  a,  ct  —  a  respectively. 
Hence  as  waves  travel  from  the  initial  region  of  disturbance  with  velocity 
c  they  leave  a  residual  disturbance  behind. 

The  essential  difference  between  the  two  cases  may  be  attributed  to 
the  fact  that  in  the  three-dimensional  case  the  wave-function  for  a  source 
is  of  type  r~lf  (t  —  r/c),  while  in  the  two-dimensional  case  it  is  of  type* 


CJ  Jo    L      c 

This  statement  may  be  given  a  physical  meaning  by  regarding  the  wave- 
function  as  the  velocity  potential  for  sound  waves  in  a  homogeneous 
atmosphere,  a  source  being  a  small  spherical  surface  which  is  pulsating 
uniformly  in  a  radial  direction. 

If  /(0  =  0,     (t<T0) 

=  1,     (Tl>t>T0) 
=  0,     (t>TJ, 

we  have  /  \t  —  -cosh  a  \  da  =  0,     (ct  <  cTQ  +  />) 

Jo    L       c  J 

-  cosh-1  [c  (t  -  T0)/p]  ,     (cT0  +  p  <  ct  <  cTl  +  p) 

-  cosh-*  [c  (t  -  T0)//>]  -  cosh-i  [c  (t  -  Tj/p], 

(cTQ  -h  p  <  ct,     cTi  +  p  <  ct). 

*  Cf.  H.  Lamb,  Hydrodynamics,  2nd  ed.  p.  474. 


Helmholtz's  Formula  189 

EXAMPLES 

1.  A  wave-function  u  is  required  to  satisfy  the  following  initial  conditions  for  t  =»  0 

u=f(x,  y),    -^  =  0    when  2  =  0, 
u  =  o,  ™  =  0     when  z  ^  0. 

C/I 

Prove  that  u  is  zero  when  z2  >  c2t2  and  when  z2  <  czt2  u  =  /  where  /denotes  the  mean  value 
of  the  function  /  round  that  circle  in  the  plane  2  =  0  whose  points  are  at  a  distance  ct  from 
the  point  (x,  y,  2). 

2.  If  in  Ex.  1  the  plane  z  =  0  is  replaced  by  the  sphere  r  =  a,  where  r2  =  x2  +  y2  -f  z2, 
the  wave-  function  w  is  equal  to  -  /  when  there  is  a  circle  (on  the  sphere)  whose  points  are  all 
at  distance  ct  from  (x,  y,  z)  and  is  otherwise  zero. 

§  2-63.  Helmholtz's  formula.  When  a  wave-function  is  a  periodic 
function  of  t,  Kirehhoff's  formula  may  be  replaced  by  the  simpler  formula 
of  Helmholtz. 

Putting  u  =  U  (x,  y,  z)  elkct 

the  wave  -equation  gives 

V2C7  +  k*U  =  0. 

Applying  Green's  theorem  to  the  space  bounded  by  a  surface  S  and  a 
small  sphere  surrounding  the  point  (xly  yly  zj  we  obtain  formula  (A) 

477  U  (x^y^zj  =-*/  (*,  y,  *)       (R~le~lkR)  dS  +  fl-ic-**  dS, 


where  R*  =  (x  -  xtf  +  (y  -  yj*  +  (z  -  ^)2, 

and  the  normal  is  supposed  to  be  drawn  out  of  the  space  under  considera- 
tion. This  space  can  extend  to  infinity  and  the  theorem  still  holds  provided 
U  -^  0  like  Ar~le~lkr  as  r->  oo,  r  being  the  distance  of  the  point  (x,  y,  z) 
from  the  origin.  It  is  permissible,  of  course,  for  U  to  become  zero  more 
rapidly  than  this. 

A  solution  of  the  more  general  equation 

V2U  +  k*U  +  a>  (x,  y,  z)  -  0 
is  obtained  by  adding  the  term 


f  1 1  R-le~lkR  a>  (x,  y,  z)  dxdydz 


to  the  right-hand  side  of  (A)  and  it  is  chiefly  in  this  case  that  we  want  to 
integrate  over  all  space  and  obtain  a  formula  in  which  U  (xl9  yi,  zj  is 
represented  by  this  last  integral. 

In  the  two-dimensional  case  when  u  is  independent  of  z,  the  function 
to  be  used  in  place  of  R-le~lkR  is  derived  from  the  function 


*u=     fit —  -cosh  a)  da, 
Jo    \       c  / 


190    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

already  mentioned.  Writing  u  =  Ueikct  as  before,  the  elementary  potential 
function  satisfying  V2U  +  k2U  =  0  is  K0  (ikp),  where  K0  (ikp)  is  defined 
by  the  equation 

K0(ikp)  =      e-ikKx»**da. 

Jo 

This  is  a  function  associated  with  the  Bessel  functions.  For  large  values 
of  R  we  have  .  .  , 


while  for  small  values  of  R 

KQ  (iR)  +  log  (B/2) 
is  finite.  The  two-dimensional  form  of  Green's  theorem  gives 

......  (B) 

where  p2  =  (x  -  a^)2  +  (y  -  2/i)2, 

rf<§  is  an  element  of  the  boundary  curve  and  n  denotes  a  normal  drawn  into 
the  region  in  which  the  point  (xl9  y^  is  situated. 
A  solution  of  the  more  general  equation 

V2U  +  k*U  +  w  (x,  y)  =  0 
is  likewise  obtained  by  adding  the  term 


—  J  J  K0  (ikp)  w(x,  y)  dxdy 


to  the  right-hand  side  of  (B).  When  k  =  0  the  corresponding  theorem  is 
that  a  solution  of  the  equation 

V2t7  +  eu  (x,  y)  =  0 
is  given  by  2-rrU  =  -    I  logpoj  (x,  y)  dxdy. 

§  2-64.    Volterra's  method*.  Let  us  consider  the  two-dimensional  wave- 
equation  r  3««     9«»     3»«      „ 

LWs-fc-te*-W~f(Xl*'Z) 

in  which  2  =  ct.  If  the  problem  is  to  determine  the  value  of  u  at  an 
arbitrary  point  (£,  77,  £)  from  a  knowledge  of  the  values  of  -u  and  ite 
derivatives  at  points  of  a  surface  /S,  we  write 

Z=*s-£     r  =  2/-7?,     Z  =  z-f, 

and  construct  the  characteristic  cone  X2  +  Y2  =  Z2  with  its  vertex  at  the 
point  (£,  77,  ^).  We  shall  denote  this  point  by  P  and  the  cone  by  the 
symbol  F. 

*  Ada  Math.  t.  xvm,  p.  161  (1894);  Proc.  London  Math.  Soc.  (2),  vol.  u,  p.  327  (1904); 
Lectures  at  Clark  University,  p.  38  (1912). 


Volterra's  Method  191 

Volterra's  method  is  based  on  the  fact  that  there  is  a  solution  of  the 
wave-equation  which  depends  only  on  the  quantity  ZjRy  where 

R2  =  X2  +  72. 

This  solution,  v,  may,  moreover,  be  chosen  so  that  it  is  zero  on  the  charac- 
teristic cone  F.  The  solution  may  be  found  by  integrating  the  fundamental 
solution  (Z2  —  X2  —  Y2)~*  with  respect  to  Z  and  is  cosh-1  w  where 
w  =  Z/R.  Since  w  =  1  on  F  it  is  easily  seen  that  v  =  0  on  F. 

For  this  wave-equation  the  directions  of  the  normal  n  and  the  co- 
normal  v  are  connected  by  the  equations 

cos  (vx)  —  cos  (nx),    cos  (vy)  =  cos  (ny),  »•  cos  (vz)  —  —  cos  (nz). 

At  points  of  F  the  conormal  is  tangential  to  the  surface  and  since  v  is 

^77 
zero  on  F,  ~-  is  also  zero.  The  function  v  is  infinite,  however,  when  R  =  0 

GV 

and  a  portion  of  this  line  lies  in  the  region  bounded  by  the  cone  F  and 
the  surface  S.  We  shall  exclude  this  line  from  our  region  of  integration 
by  means  of  a  cylinder  (7,  of  radius  c,  whose  axis  is  the  line  R  =  0.  We 
now  apply  the  appropriate  form  of  Green's  theorem,  which  is 


to  the  region  outside  C  and  within  the  realm  bounded  l->y  S  and  F.  On 
account  of  the  equations  satisfied  by  u  and  v  the  forgoing  equation 
reduces  simply  to 

JU-  *-*)"--  JJJ*fc- 

On  C  we  have 


dS  = 
and  since  lim  (s  log  e)  =  0 

e  ->0 

we  have  Hm  j  j^  (u  |  -  v^)  -  -  2»  |*  «  tf  ,  ,,  «)  &, 

where  (^,  r\>  z)  is  on  A$f  and  is  in  the  part  of  8  excluded  by  C.   We  thus 
obtain  the  formula 


and  the  value  of  u  at  P  may  be  derived  from  this  formula  by  differentiating 
with  respect  to  £.  The  result  is 


192     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 


EXAMPLE 

Prove  that  a  solution  of  the  equation 


dx*  +  By*  =  c2  W 
is  given  by  the  following  generalisation  of  Kirchhoff's  formula, 

2nu  (x,  y,t)  =  /    [c2  (t  -  tj*  -  r2]"*  {cos  nt  -  cos  nr  .  c  (t  -  tj/r}  u  fa,  ft  ,  fj  dSl 

J  <r 


f  f 


in  which  r2  =  (x  —  a^)2  +  (y  —  ft)2  and  the  integration  extends  over  the  area  a  cut  out  on 
a  surface  8  in  the  (x19yl9t1)  space  by  the  characteristic  cone 

fa  -  *)2  +  (ft  -  y)2  =  c2  &  -  t)\ 

the  time  t  being  chosen  so  as  to  satisfy  the  inequality  t  <  t±  . 

[V.  Voltorra.] 

§  2-71.  Integral  equations  of  electromagnetism.  Let  us  consider  a  region 
of  space  in  which  for  some  range  of  values  of  t  the  components  of  the 
field-vectors  E  and  H  and  their  first  derivatives  are  continuous  functions 
of  x,  y,  z  and  t. 

Take  a  closed  surface  S  in  this  region  and  assign  a  time  t  to  each  point 
of  S  and  the  enclosed  space  in  accordance  with  some  arbitrary  law 

t  =  f(z,  y,  z), 

where  /  is  a  function  with  continuous  first  derivatives.  We  shall  suppose 
that  this  function  gives  for  the  chosen  region  a  value  of  t  lying  within  the 
assigned  range  and  shall  use  the  symbol  T  to  denote  the  'vector  with 

4.         3/         3/         9/ 

components^,  0^. 

Writing  Q  =  H  +  iE  as  before  we  consider  the  integral 


/  =  Jj  [Q  -  i  (Q  x  T)]n  dS 


taken  over  the  closed  surface  S.  The  suffix  n  indicates  the  component  along 
an  outward  -drawn  normal  of  the  vector  P  which  is  represented  by  the 
expression  within  square  brackets.  Transforming  the  surface  integral  into 
a  volume  integral  we  use  the  symbol  Div  P  to  denote  the  complete  diver- 
gence when  the  fact  is  taken  into  consideration  that  P  depends  upon  a 
time  t  which  is  itself  a  function  of  x,  y  and  z.  The  symbol  div  Q,  on  the 
other  hand,  is  used  to  denote  the  partial  divergence  when  the  fact  that 
Q  depends  upon  x,  y  and  z  through  its  dependence  on  t  is  ignored.  We 
then  have  the  equation 

/  =  1  1  j  div  P  .  dr, 


where  div  P  -  div  Q  +  T .  R, 

3Q 

and  R  -  -57  —  i  curl  Q. 

ot 


Integral  Equations  of  Electromagnetism  193 

Now  div  Q  and  R  vanish  on  account  of  the  electromagnetic  equations 
and  so  these  equations  are  expressed  by  the  single  equation  /  =  0.  When 
/  is  constant  T  =  0  and  the  equation  7=0  gives 


which  correspond  to  Gauss's  theorem  in  magneto-  and  electrostatics.  It 
may  be  recalled  that  Gauss's  theorem  is  a  direct  consequence  of  the  inverse 
square  law  for  the  radial  electric  or  magnetic  field  strength  due  to  an 
isolated  pole.  The  contribution  of  a  pole  of  strength  e  to  an  element  EdS 
of  the  second  integral  is,  in  fact,  eda>/4t7T,  where  cfa>  is  the  elementary  solid 
angle  subtended  by  the  surface  element  dS  at  this  pole.  On  integrating 
over  the  surface  it  is  seen  that  the  contribution  of  the  pole  to  the  whole 
integral  is  e,  \e  or  zero  according  as  the  pole  lies  within  the  surface,  on 
the  surface  or  outside  the  surface. 

This  result  is  usually  extended  to  the  case  of  a  volume  distribution  of 
electricity  by  a  method  of  summation  and  in  this  case  we  have  the  equation 


where  p  denotes  the  volume  density  of  electricity. 

Transforming  the  surface  integral  into  a  volume  integral  we  have  the 
equation 

J  j  J  (div  E  -  p)  dr  =  0, 

which  gives  div  E  =  p. 

Since  E  =  —  V</>  the  last  equation  is  equivalent  to  Poisson's  equation 

V  V  +  p  =  0, 

in  which  the  factor  4?r  is  absent  because  the  electromagnetic  equations 
have  been  written  in  terms  of  rational  units.  Our  aim  is  now  to  find  a 
suitable  generalisation  of  this  equation.  In  order  to  generalise  Gauss's 
theorem  the  natural  method  would  be  to  start  from  the  field  of  a  moving 
electric  pole  and  to  look  for  some  generalisation  of  the  idea  of  solid  angle. 
This  method,  however,  is  not  easy,  so  instead  we  shall  allow  ourselves  to 
be  guided  by  the  principle  of  the  conservation  of  electricity.  The  integral 

which  must  be  chosen  to  replace       P^T  should  be  of  such  a  nature  that 

its  different  elements  are  associated  with  different  electric  charges  when 
each  element  is  different  from  zero.  When  the  elements  are  associated 
with  a  series  of  different  positions  of  the  same  group  of  charges  which  at 
one  instant  lie  on  a  surface  it  may  be  called  degenerate.  In  this  case  we 

B  13 


194    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

can  regard  these  charges  as  having  a  zero  sum  since  a  surface  is  of  no 
thickness.  Now  it  should  be  noticed  that  if  we  write 

0  =  *-/(*,*/,  z), 
the  quantity 

M     SO  ,       W  (       80  ,       W      _      ,     ^ 

:/;  =  27  +  vx  %-  +  vv  5-  +  v«  o-  =  1  —  (0  •  T) 
dt      dt          ox        vcy       z  dz  ' 

vanishes  when  the  particles  of  electricity  move  so  as  to  keep  6  =  0,  that 
is  so  as  to  maintain  the  relation  t  =  /  (xy  y,  z),  and  in  this  case  the  integral 

dd 


is  degenerate. 

We  shall  try  then  the  following  generalisation  of  Gauss's  theorem  and 
examine  its  consequences  : 

I=i\\\P[l-(v.T)]dT. 


Transforming  the  surface  integral  into  a  volume  integral  we  have 
0  =        [div  Q  -  ip  +  T  .  (R  4-  ipv)]  dry 


and  since  the  function  /  is  arbitrary  this  equation  gives 

div  Q  =  ip,     R  =  -  ipv. 
Separating  the  real  and  imaginary  parts  we  obtain  the  equations 

3E 
curl  H  =  -fa  +  />v,     div  E  =  p, 

curlE=  -  I?,        divH=0, 
ct 

which  are  the  fundamental  equations  of  the  theory  of  electrons.  The  first 
two  equations  give  ~ 


which  is  analogous  to  the  equation  of  continuity  in  hydrodynamics.  Our 
hypothesis  is  compatible,  then,  with  the  principle  of  the  conservation  of 
electricity.  The  integral  equation 

[Q-  »(Q  x  T)]ndS=  »}J|p[l  -  (v.T)]dr  (A) 

will  be  regarded  as  more  fundamental  than  the  differential  equations  of 
the  theory  of  electrons  if  the  volume  integral  is  interpreted  as  the  total 
charge  associated  with  the  volume  and  is  replaced  by  a  summation  when 
.the  charges  are  discrete.  This  fundamental  equation  may  be  used  to  obtain 
the  boundary  conditions  to  be  satisfied  at  a  moving  surface  of  discontinuity 
which  does  not  carry  electric  charges. 

Let  t  =  f  (x,  y,  z)  be  the  equation  of  the  moving  surface  and  let  the 


Boundary  Conditions  195 

surface  8  be  a  thin  biscuit-shaped  surface  surrounding  a  superficial  cap  S0 
at  points  of  which  t  is  assigned  according  to  the  law  t  =/(#,  y,  z).  At 
points  of  the  surface  S  we  shall  suppose  t  to  be  assigned  by  a  slightly 
different  law  t  =  /x  (#,  t/,  z)  which  is  chosen  in  such  a  way  that  the  points 
of  S  on  one  face  have  just  not  been  reached  by  the  moving  surface 
t  =  f  (x,  y,  z),  while  the  points  on  the  other  face  have  just  been  passed  over 
by  this  surface.  Taking  the  areas  of  these  faces  to  be  small  and  the  thick- 
ness of  the  biscuit  quite  negligible  the  equation  (A)  gives 

[Q'  -  •  (Q'  x  T)]»  =  [Q"  -  i  (Q"  x  T)]n, 

where  Q',  Q"  are  the  values  of  Q  on  the  two  sides  of  the  surface  of  dis- 
continuity and  the  difference  between  /x  and  /  has  been  ignored.  Writing 
q  =  Q'  —  Q"  we  have  the  equation 

[q  -  *  (q  x  T)]n  =  0. 
Now  the  direction  of  the  cap  $0  is  arbitrary  and  so  q  must  satisfy  the 

relation  Q-.'(qxT). 

This  gives  q2  =  0.   Hence  if  q  =  h  +  ie  we  have  the  relations 

h*  -  &  =  0,     (h.e)  =  0. 
The  equation  also  gives  (q  .  T)  =  0, 

and  (q  x  T)  =  i  (q  x  T)  x  T  -  i  [T  (q  .  T)  -  qT2] 

=  -  iT2q     or  T2  =  1     if  q  ^  0. 

Hence  the  moving  surface  travels  with  the  velocity  of  light. 

A  similar  method  may  be  used  to  find  the  boundary  conditions  at  the 
surface  of  separation  between  two  different  media.  We  shall  suppose  that 
the  media  are  dielectrics  whose  physical  properties  are  in  each  case  specified 
by  a  dielectric  constant  K  and  a  magnetic  permeability  /*.  For  such  a 
medium  Maxwell's  equations  are 

,        divD-0, 


^-— 

where  D  =  #E,    B 

Instead  of  these  equations  we   may  adopt  the  more  fundamental 
integral  equations 


J| 


[B  +  (E  : 

which  give  the  generalisations  of  Gauss's  theorem.  The  boundary  con- 
ditions derived  from  these  equations  by  the  foregoing  method  are 

d  -  (h  x  T)  =  0,    b  +  (e  x  T)  =  0, 

13-2 


196     Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

where  e,  h,  d,  b  are  the  differences  between  the  two  values  of  the  vectors 
E,  H,  D,  B  respectively  on  the  two  sides  of  the  moving  surface.  These 
equations  give  (A.T)  =  0,  (b.T)^0, 

(d  x,  T)  +  T*h  =  (h .  T)T;     (b  x  T)  -  T2e  -  -  (e  .  T)  T. 
If  the  vector  v  represents  the  velocity  along  the  normal  of  the  moving 
surface  we  have  v^  ^  y      VT  =  \ 

hence  the  equations  may  be  written  in  the  form 

dv  =  0,     bv  =  0,     hr  =  (v  y  d)r,     eT  =  —  (v  x  b)T, 

where  dv ,  bv  denote  components  of  d  and  6  normal  to  the  moving  surface 
and  the  suffix  r  is  used  to  denote  a  component  in  any  direction  tangential 
to  the  moving  surface.  When  this  surface  is  stationary  the  conditions  take 
the  simple  form 

dv  =  0,     b¥  =  0,     hr  =  0,     er  =  0 

used  by  Maxwell,  Rayleigb  and  Lorentz. 

When  a  surface  of  discontinuity  moves  in  a  medium  with  the  physical 
constants  K  and  /z,  we  have  Heaviside's  equations  (Electrical  Papers, 
vol.  ir,  p.  405)  X(e.T)  =  0,  /*<h.T)  =  0, 

K  (e  x  T)  +  T2h  -  0,    p.  (h  x  T)  -  T2e  =  0,. 
and  so  A>  [(h  x  T)  x  T]  =  KT*  (e  x  T)  -  -  T4h, 

i.e.  Kfji  -  T2 

if  h  ^  0. 

The  surface  thus  moves  with  a  velocity  v  given  by  the  equation 


§  2-72.    The  retarded  potentials  of  electromagnetic  theory.  The  electron 

equations  ,    ~™ 

i  [  (j  ty          \ 

curl  H  =  -  {  ~     +  pv),     div  E  =  p, 

C  \  ut  I 

curl  E  =  -      -37,  div  //  =  0 

c  3£ 

may  be  satisfied  by  writing 

c   o£ 
where  the  potentials  A  and  O  satisfy  the  relations 


Retarded  Potentials  197 

The  last  equations  are  of  the  type  to  which  Kirchhoff  's  formula  is  applicable 
and  so  we  may  write 

......  (B) 


These  are  the  retarded  potentials  of  L.  Lorenz. 

The  corresponding  potentials  for  a  moving  electric  pole  were  obtained 
by  Lienard  and  Wiechert.  They  are  similar  to  the  above  potentials  except 
that  the  quantity  —  c/M  of  §  1-93  takes  the  place  of  1/r.  Let  £  (£),  ^  (t), 
£  (t)  be  the  co-ordinates  of  the  electric  pole  at  time  t  and  let  a  time  r  be 
associated  with  the  space-time  point  (x,  y,  z,  t)  by  means  of  the  relations 

[X  -  t  (T)]«  +[y-r,  (T)]«  +  [z  -  £  (T)]«  =  c*  (t  -  T)«,      r  <  t,     (C) 
then 

M=[x-t  (r)}  ?  (r)  +  [y-r,  (r)]r,'  (r)  +  [z  -  I  (r)]  £'  (r)  -  c'  («  -  r), 
and  if  e  is  the  electric  charge  associated  with  the  pole  the  expressions  for 
the  potentials  are  respectively 

A    --*>(?\        A    -_?9l<T)       A    -_<»       *-_     e-C 
x~  ''  ~  '         z~  ' 


These  satisfy  the  relation  (A)  and  give  the  formulae  of  Hargreaves 

,-,  e     d  (or,  T)        „    _    e    9  (a,  r) 

x  =  4^c  3  (x,  0  '        *  =  ITT  3ly,  z)  ' 
where 

o  =  [*  -  f  (r)]  f'  (r)  +  [y  -  i,  (T)]  ,"  (T)  +  [z  -  £  (T)]  C"  (T) 

+  C*  -  [£'  (T)]«  -  [,'  (T)]»  -  [£'  (T)]«. 

It  should  be  remarked  that  the  retarded  potentials  (B)  can  be  derived 
from  the  Lienard  potentials  by  a  process  of  integration  analogous  to  that 
by  which  the  potential  function 


==  ttl 


is  derived  from  the  potential  of  an  electric  pole. 

Instead  of  considering  each  electric  pole  within  a  small  element  of 
volume  at  its  own  retarded  time  r  we  wish  to  consider  all  these  electric 
poles  at  the  same  retarded  time  TO  belonging,  say,  to  some  particular  pole 
(&>  >?o>  £o,  TO)-  Writing 

f  W  =  £>  (a)  +  a  (a),     rj  (a)  =  rjQ  (a)  +  ft  (a),     £  (or)  =  £0  (a)  +  y  (a), 
where  a  (a),  ft  (a),  y  (a)  are  small  quantities,  we  find  that  if  T  is  defined  by 
(C), 
(r  -  T0)  [(x  -  &)  &  +(y-  %)  V  +  (2  -  Co)  &,'  -  c«  («  -  T0)] 

+  (*-&)«+(»-  %)  ^  +  (z  -  £0)  7  =  0, 

where  £0,  ij0,  £0,  ^0'>  ^o'>  £o'>  a>  /S>  y  are  all  calculated  in  this  equation  at 
time  TO. 


198    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

On  account  of  the  motion  the  pole  (£,  77,  £)  occupies  at  time  r  the 
position  given  by  the  co-ordinates 

£  =  fo  +  «  +  (r  -  T0)  &', 

*?  =  i?o  +  j8  +  (T  -  T0)  V.  >a 

£  =  £o  +  y  +  (r  -  T0)  £0'. 

-1^'* 


If  p  is  the  density  of  electricity  when  each  particle  in  an  element  of  volume 
is  considered  at  the  associated  time  T  and  p0  is  the  density  when  each 
particle  is  considered  at  time  TO,  we  have 

pd  (f,  T?,  £)  -  Pod  (a,  j8,  y). 

C7" 

Therefore  o0  ==  p        —  --  , 

ru      r  cr  —  (r  .  v) 

and  so  f  I  f  ^  dxdydz  -  f  f  f      -^-     ,  dxdydz. 

c  JJJ      ^  JJJ^-(^.V) 

Writing  p  dxdydz  ^  de  we  obtain  the  Lienard  potentials. 

Similar  analysis  may  be  used  to  find  the  field  of  a  dipole  which  moves 
in  an  arbitrary  manner  with  a  velocity  less  than  c  and  at  the  same  time 
changes  its  moment  both  in  magnitude  and  direction. 

Let  us  consider  two  electric  poles  which  move  along  the  two  neigh- 
bouring curves 

*  =  £(*),     y-^W,     *  =  £(0> 

x  =  f(t)+€a  (t),     y  =  TJ  (*)  +  cjB  (t),     z  =  J  (0  +  ey  (0, 
e  being  a  quantity  whose  square  may  be  neglected.  If  TX  is  defined  in  terms 
of  xy  y,  Zy  t  by  the  equation 


and  T!  =  T  +  €0,  we  easily  find  that 

JM  +  a  (T)  [x-£  (T)]  +  ft  (r)  [y  _  ,  (T)]  +  y  (T)  [z  _  J  (T)]  =  Q. 

If  3/x  is  the  quantity  corresponding  to  M  ,  we  have 

J^  =  Jf  +  c  [^^CT  +  (x  -  f)a'  +  (y  -  T?)  $  -f  (^  -  t)  /  -  «f  -  fa'  -  y£'] 

=  Jlf  -f  €[&Jf<7  +  p], 

say,  where  a  has  the  same  meaning  as  before  and  primes  denote  differentia- 
tions with  respect  to  T.  Now  if 

_  eff'frJ  +  ^Kll      ,.,,_          ec 

'  ~ 


-_ 

477J/' 


Moving  Electric  and  Magnetic  Dipoles  199 

we  have 

ax  =     [A.'  -Ax}=-          ,  [Ma'  -  p?  +  MB?'  -  MOaf], 


But  Ma!  -  p?  +  M6("  -  M6a£'  s  (y  -  r,)  n'  -  (z  -  Q  m'  -  c2  (t  -  T)  «' 

+  c2«  -  nrf  +  m£'  +  o{a(t-  T)  —  n(y  —  ij)  +  m(z  —  £)}» 
where  I  =  ft'  -  m',    m  =*  y?  -  oj',     »  =  <nj'  -  #'. 

Hence  we  may  write 

_    e  riL^7^  _  i  f!M  0.1  f-^i 

a*  ~  ~  47r  [fy  \M)      dz  \MJ  +  9<  \M)\  ' 

ecp  /«\    a  //?\    a/y\i 
^  ~  4^  [ai  UJ  +  %  \M)  +  dz  \M)\  • 

These  results  may  be  obtained  also  with  the  aid  of  the  general  theorem 
which  gives  the  effect  of  an  operation  -r  analogous  to  differentiation, 

i  r  i  dn  =  i  n  ^ 


i  dn  =  i  n  ^DI  ,  i  n  i 

f  drj      ao;  [M  d  (e,  r)  J  "  dy  lMd(€,r 


/  being  a  function  of  r  and  6.    Writing  /  =  f  ,  ^-  =  a,  =-'  =  /J,  ^-  =  y  the 

expression  for  a^  is  at  once  obtained  from  that  for  ^4X.   Writing/  =  r  we 
obtain  the  expression  for  (f>. 

The  formulae  show  that  the  field  of  the  moving  dipole  may  be  derived 
from  Hertzian  vectors  II  and  F  by  means  of  the  formulae 


where  u  and  w  are  vector  functions  of  r  with  components  (a,  j3,  y),  (Z,  m,  n) 
respectively.  If  v  denotes  the  vector  with  components  (£',  T\  ',  £x)  we  have 
the  relations  (v  .  w)  =  0,  (u  .  W)  -  0, 

consequently  Hertzian  vectors  of  types  (D)  do  not  specify  the  field  of  a 
moving  electric  dipole  unless  these  relations  are  satisfied. 

Since  A  =  -  37  +  curl  T,     B  =  -  ^  -  curl  II, 

c  ot  c  ot 


where  B  and  O  are  the  electromagnetic  potentials  of  magnetic  type,  we 
may  write  down  the  potentials  for  a  moving  magnetic  dipole  by  analogy. 


200    Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

We  simply  replace  II  by  T  and  F  by  —  II.  Hence  the  potentials  of  a  moving 
magnetic  dipole  are  of  type 


a    -  -  -- 

x  ~  4n    dy  \M)      dz  \M)      dt  \M 

me  f"  3  /  /  \       9  fm\      d  /  n 
~ 


Let  us  now  calculate  the  rate  of  radiation  from  a  stationary  electric 
dipole  whose  moment  varies  periodically.  Taking  the  origin  at  the  centre 
of  the  dipole,  we  write 

HX  =  ^/(T),     Hv=±g(T)9     ilz  =  -rh(r),     r=«-r/c, 

where  /,  y,  h  are  periodic  functions  with  period  T.  The  vector  is  zero  since 
there  is  no  velocity. 

In  calculating  the  radiation  we  need  only  retain  terms  of  order  1/r  in 
the  expressions  for  E  and  H  .  To  this  order  of  approximation  we  have 


=-  (yE,  -  zEy), 


where  ^  =  ^  (/"*  +  g"*  +  h"*)  -  ~  (xf"  +  yg"  +  zh")*. 

The  rate  of  radiation  is  obtained  by  integrating  cE2  over  a  spherical 
surface  r  =  a,  where  a  is  very  large.  With  a  suitable  choice  of  the  axis 
of  z  we  may  write 


and  the  value  of  the  integral  over  the  sphere  is 


the  mean  value  over  a  period  T  is 

A2       /27T\4 

"  " 


Electromagnetic  Radiation  201 

EXAMPLE 

and         L  (x,  y,  z,  t,  s)  =  [x  -  £  (s)]  I  (s)  +  [y  —  ^  (5)]  m  (*)  -f  [2  -  £  («*)]  n  (s)  -  c(t  ~  s), 
prove  that  the  potentials 

1    iT     ft  i  \       l(s)ds  v£'(T) 

*  =  4w  J  -  oo          £  (#,  y,  2,  /,~«)  "  ' (     4^ ' 

4    _  1    fr      ///^       mW^ 


/     ^' £(*,*,,*,*,*)       'V''4^' 

are  wave-functions  satisfying  the  condition 

1V      "*"  c  dt  =    * 

Show  also  that  in  the  field  derived  from  these  potentials  the  charge  associated  with  the 
moving  point  £(T),  y  (T),  £(T)  is/(r),  the  variation  with  the  time  being  caused  by  the 
radiation  of  electric  charges  from  the  moving  singularity  in  a  varying  direction  specified  by 
the  direction  cosines  /  (r),  m  (T),  n  (T). 

§  2-73.  The  reciprocal  theorem  of  wireless  telegraphy.  If  we  multiply  the 
electromagnetic  equations 

JSj  =  —  curl  (cEi),     C1  =  curl  (c^) 

for  a  field  (Ely  H^  by  H2,  —  E2  respectively,  where  (E2,  H2)  are  the  field 
vectors  of  a  second  field  in  the  same  medium,  and  multiply  the  field 

equations  A  ,  ,  ^  x       ~  i  /  r?  x 

^  B2  =  —  curl  (cE2),     C2  =  curl  (cH2) 

for  this  second  field  by  —  H19  El  respectively  and  then  add  all  our  equations 

together,  we  obtain  an  equation  which  may  be  written  in  the  form 

(H2 .  B,)  -  (H, .  B2)  +  (El .  C2)  -  (E2 .  C,)  =  cdiv  (E2  x  HJ  -  cdiv  (EI  x  H2). 

(A) 

We  now  assume  that  both  fields  are  periodic  and  have  the  same  frequency 
co/277.  Introducing  the  symbol  T  for  the  time  factor  e~ltat  and  assuming 
that  it  is  understood  that  only  the  real  part  of  any  complex  expression 
in  an  equation  is  retained,  we  may  write 

TT        WL        u        nnjt        ~&        ^n^t        &        nn^i 
/i-i==j[/ij,.  juL2  —  j.  n2j     /vj  =  JL  6j ,     jG/2  —  ,JL  e2) 

where  the  vectors  ely  hl9  cl9  etc.  depend  only  on  x,  y  and  z. 

Now  let  K,  fji  and  a  be  the  specific  inductive  capacity,  permeability  and 
conductivity  of  the  medium  at  the  point  (x,  y,  z)  and  let  a  denote  the 
quantity  (ioo  -f  icr)/c,  then  we  have  the  equations 

B2  ==  —  ia)jjih2T,     Cl  =  —  iceiTa,    C2  =  —  ice2Ta, 


202   Applications  of  the  Integral  Theorems  of  Gauss  and  Stokes 

which  indicate  that  the  left-hand  side  of  equation  (A)  vanishes.  The 
equation  thus  reduces  to  the  simple  form  * 

div  (e2  x  hj)  =  div  (e1  x  h2). 

This  equation  may  be  supposed  to  hold  for  the  whole  of  the  medium 
surrounding  two  antennae  f  if  these  sources  of  radiation  are  excluded  by 
small  spheres  Kl  and  K2  .  An  application  of  Green's  theorem  then  gives 

[   (%  x  ^i)n  dS  -f  [   (e2x  hi)n  dS  -  I    (el  x  h2)n  dS  +  [   (el  x  h2)n  dS, 

J  KI  J  Kt  J  KL  J  KZ 

an  equation  which  may  be  written  briefly  in  the  form 

Jn  -f-  J21  —   J12  4-  </22- 

Let  the  first  antenna  be  at  the  origin  of  co-ordinates  and  let  us  suppose 
for  simplicity  that  it  is  an  electrical  antenna  whose  radiation  may  be 
represented  approximately  in  the  immediate  neighbourhood  of  O  by  the 
field  derived  from  a  Hertzian  vector  TIT  with  a  single  component  HZT, 
where 


=  MelpR     (c2p2  =  kfjLa>2  -f 

and  M  is  the  moment  of  the  dipole.  In  making  this  assumption  we  assume 
that  the  primary  action  of  the  source  preponderates  over  the  secondary 
actions  arising  from  waves  reflected  or  diffracted  by  the  homogeneities  of 
the  surrounding  medium. 

Using  %  to  denote  the  value  of  a  at  0  and  writing  FT  for  3FI/9r,  (£2  >  y*  >  £2) 
for  the  components  of  e2  ,  we  have 


n  =  -  ia,  f  {(** 

J  K, 


-f  y2)  £2  -  a*&  -  yz^}  H'R~*dS. 


For  the  integration  over  the  surface  Kl  the  quantities  7?2,  Ilx  and  the 
vector  e2  may  be  treated  as  constants,  for  K1  is  very  small  and  e2  varies 
continuously  in  the  neighbourhood  of  0.  We  also  have 


[yzdS  -  \zxdS  =  \xydS  =  0,     (x*dS=  ^dS=  \z2dS  = 
IxdS^O,     lx*dS  =  0,     l 


Therefore  Jn  =  - 


and,  since  lim 

.R->0 

we  have  finally  Ju  =  2ia1M1£2/3. 

Now  the  rate  at  which  the  antenna  at  0  radiates  energy  is 
8  =  ^W2127rF3,         V  =  c  (€x)'i. 


*  H.  A.  Lorentz,  Amsterdam.  Akad.  vol.  iv,  p.  176  (1895-6). 

f  A.  Sommerfeld,  Jahrb.  d.  drahtl  Telegraphie,  Bd.  xxvi,  S.  93  (1925);  W.  Schottky,  ibid. 
Bd.  xxvn,  S.  131  (1926). 


Reciprocal  Relations  203 

Assuming  that  8  is  the  same  for  both  antennae  we  obtain  the  useful 
expression 


The  integral  J21  is  seen  to  be  zero  because  it  involves  only  terms  which 
change  sign  when  the  signs  of  x,  y  and  z  are  changed. 

Evaluating  «/22  and  Ju  in  a  similar  way  we  obtain  the  equation 

(^  +  icrjaj)  (VtfK^  £2  =  (ic,  +  iaju)  (F23//c2)i  &, 

where  f1?  rjl9  £x  are  the  components,  at  the  second  antenna  02,  of  the  vector 
elf  The  amplitudes  and  phases  of  the  field  strength  received  at  the  two 
antennae  are  thus  the  same  when  both  antennae  are  of  the  electric  type 
and  are  situated  at  places  where  the  medium  has  the  same  properties  and 
emits  energy  at  the  same  rate.  When  the  two  antennae  are  both  of  magnetic 
type  the  corresponding  relation  is 

(M2F23)*ri=(Mi^3)*y2, 

where  a^  ,  &  ,  y^  are  the  components  of  ht  and  a2  ,  /J2  ,  y2  are  the  components 
of  A2.  The  antennae  are  again  supposed  to  be  directed  along  the  axis  of 
z  but  there  is  a  more  general  theorem  in  which  the  two  antennae  have 
arbitrary  directions. 

The  relation  (A)  and  the  associated  reciprocal  relation  remind  one 
of  the  very  general  extension  of  Green's  theorem  which  was  given  by 
Volterra*  for  the  case  of  a  set  of  partial  differential  equations  associated 
with  a  variational  principle.  This  extension  of  Green's  theorem  is  closely 
connected  with  a  property  of  self-adjointness  which  has  been  shown  by 
Hirsch,  Kiirsch&k,  Davis  and  La  Pazj  to  be  characteristic  of  certain  equa- 
tions associated  with  a  variational  principle.  In  the  case  of  the  Eulerian 
equation  F  =  0  associated  with  a  variational  principle  87  =  0,  where 

l>i,32,  •••  Xn\u\  ultu2j  ...  un]dxldx2  ...dxn, 

du  d*u  .  1    0 

Us  =  7^  >     u"  =  ^T^r>         (r>  s  =  !>  2>  •••  n)> 

x  UJsg  ,       CJCr  VJUg 

the  equation  which  is  self  -adjoint  is  the  "equation  of  variation"  for  v 

dF  3F  dF  dF  3F 

0  =  v  =  --  h  ^  -A  --  h  ...  vn  ~  --  h  vn  --  ---  h  v12  x-  -  +  ...  . 

du         1  dut  dun  dun        li  du12 

*  V.  Volterra,  Rend.  Lincei  (4),  t.  vi,  p.  43  (1890). 

f  A.  Hirsch,  Math.  Ann.  Bd.  XLIX,  S.  49  (1897);  J.  Kurschdk,  ibid.  Bd.  LX,  S.  157  (1905); 
D.  R.  Davis,  Trans.  Amer.  Math.  Soc.  vol.  xxx,  p.  710  (1928);  L.  La  Paz,  ibid.  vol.  xxxir,  p.  509 
(1930). 


CHAPTER          ^ 
TWO-DIMENSIONAL   PROBLEMS 

§  3-11.  Simple  solutions  and  methods  of  generalisation  of  solutions.  A 
simple  solution  of  a  linear  partial  differential  equation  of  the  homo- 
geneous type  is  one  which  can  be  expressed  in  the  form  of  a  product  of  a 
number  of  functions  each  of  which  has  one  of  the  independent  variables 
as  its  argument.  Thus  Laplace's  equation 

927 


== 

dx2      dy2 
possesses  a  simple  solution  of  type 

V  =  e~my  cos  m(x  -  x'),  ......  (A) 

where  m  and  x'  are  arbitrary  constants  ;  the  equation 

9F_     d2V 
~ft-K~dx* 

ppssesses  the  simple  solution 

V  =  e~Km2t  cos  m  (x  -  x'},  ......  (B) 

and  the  wave-equation 


^v 
dx2  ^  c2 


possesses  the  simple  solution 


V  =  —  sin  met  cos  m  (x  —  x').  ......  (C)     *• 

itv 

The  last  one  is  of  great  historical  interest  because  it  was  used  by  Brook 
Taylor  in  a  discussion  of  the  transverse  vibrations  of  a  fine  string.  It 
should  be  noticed  that  the  end  conditions  F  =  0  when  x  =  ±  a/2  are 
satisfied  by  a  solution  of  this  type  only  if  ma  =  2n  -f  1,  where  n  is  an 
integer.  There  are  thus  periodic  solutions  of  period 

T  =  27r/wc  -  2ira/(2n  -f  1). 

If  M  (m.,  x')  denotes  one  of  these  simple  solutions  a  more  general 
solution  may  be  obtained  by  multiplying  by  an  arbitrary  function  of  m 
and  x'  and  then  summing  or  integrating  with  respect  to  the  parameters 
m  and  x'.  This  method  of  superposition  is  legitimate  because  the  partial 
differential  equations  are  linear.  When  infinite  series  and  infinite  ranges 
of  integration  are  used  it  is  not  quite  evident  that  the  resulting  expression 
will  be  a  solution  of  the  appropriate  partial  differential  equation  and  some 


Generalisation  of  Simple  Solutions  205 

process  of  verification  is  necessary.  If,  for  instance,  we  take  as  our  generalisa- 
tion the  integral 

V  =  f°°  M  (m,  x')f(m]  dm         (t  >  0,  y  >  0), 

Jo 

and  distinguish  between  solutions  of  the  different  equations  by  writing 
v  for  V  when  we  are  dealing  with  a  solution  of  the  second  equation  and 
y  for  V  when  we  are  dealing  with  a  solution  of  the  third  equation,  we  easily 
find  that  when  /  (m)  =  1  we  have 


2v  =  (77//c£)*exp  [—  (x  —  x')*l±Kt], 
y  =  - ,  -  or  0  according  as  |  x  —  x'  \  =  c£        (£  >  0). 

It  is  easily  verified  that  these  expressions  are  indeed  solutions  of  their 
respective  equations.  These  solutions  are  of  fundamental  importance 
because  each  one  has  a  simple  type  of  point  of  discontinuity.  In  the  last 
case  the  points  of  discontinuity  for  y  move  with  constant  velocity  c. 

We  may  generalise  each  of  these  particular  solutions  by  writing  V,  v 
or  y  equal  to 

M  (m,  x')  F  (xf)  dx'dm, 

Jo    J  -oo 

where  the  integration  with  regard  to  x'  precedes  that  with  respect  to  m. 
When  the  order  of  integration  can  be  changed  without  altering  the  value 
of  the  repeated  integral  the  resulting  expressions  are  respectively 

F=p°      yF(x')dx' 


2v  =  (77/Ac*)*  I""  exp  [-  (x  -  x')*l±K(\  F  (xf)  dx', 

J  -00 

—   fjr  +  ct 

y=l\       F(x')dx'. 

*  Jx-ct 

The  last  expression  evidently  satisfies  the  differential  equation  when  F  (x) 
is  a  function  with  a  continuous  derivative;  y  represents,  moreover,  a 
solution  which  satisfies  the  conditions 


when  t  =  0. 

When  the  function  F  (x)  is  of  a  suitable  type  the  functions  V  and  v  also 
satisfy  simple  boundary  conditions.  This  may  be  seen  by  writing 

x'  -  x  +  y  tan  (6/2) 
in  the  first  integral  and         x'  =*  x  +  2u  (irf)* 


206  Two-dimensional  Problems 

in  the  second.  The  resulting  classical  formulae 


Too 

v  =  TT*         F[x+2u  (AC 

J  -00 


u 


suggest  that  V  =  nF  (x)  when  t/  =  0  and  v  =  -nF  (x)  when  2=0. 

These  results  are  certainly  true  when  the  function  F  (x)  is  continuous 
and  integrable  over  the  infinite  range  but  require  careful  proof.  The 
theorems  suggest  that  in  many  cases  * 

rrF  (x)  =  [  °°  dm  f  °°  cos  m(x-  x')  F  (x')  dx'. 

Jo  J  -oo 

This  is  a  relation  of  very  great  importance  which  is  known  as  Fourier's 
integral  theorem.  Much  work  has  been  done  to  determine  the  conditions 
under  which  the  theorem  is  valid. 

A  useful  equivalent  formula  is 


(x)  =  I  °°  dm  f  °°  etm(*-*'>  F  (xf)  dx'. 

J  —oo  J  —oo 


When  F  (x)  is  an  even  function  of  x  Fourier's  integral  theorem  may  be 
replaced  by  the  reciprocal  formulae 


f  00 

F  (x)  —       cos  mxG  (m)  dm, 

Jo 

2  f  °° 

0  (m)  =-        co$mxF(x)dx, 

7T  Jo 


and  when  F  (x)  is  an  odd  function  of  x  the  theorem  may  be  replaced  by 
the  reciprocal  formulae 

f  °° 

F  (x)  =       sin  mxH  (m)  dm, 
Jo 

2  f°° 
H  (m)  =  -       sin  mxF  (x)  dx. 

TT  JO 

The  formulae  require  modification  at  a  point  x,  where  F  (x)  is  discon- 
tinuous. It  F  (x)  approaches  different  finite  values  from  different  sides  of 

*  The  theorem  is  usually  established  for  a  continuous  function  which  is  of  bounded  variation 

fee  A) 

and  is  such  that  /        |  F  (x)  \  dx  and  I        |  F  (x)  \  dx  exist.  F  (x)  may  also  have  a  finite  number  of 

/     -00  J     -JO 

points  of  discontinuity  at  which  F  (x  +  0)  and  F  (x  -  0)  exist  but  in  this  case  the  integral  represents 

'J[[F(x+  0)+  F  (x  -  0)].    Proofs  of  the  theorem  are  given  in  Carslaw's  Fourier  Series  and 

2    • 

Integrals;  in  Whittaker  and  Watson's  Modern  Analysis;  and  in  Hobson's  Functions  of  a  Eeal 

Variable. 


Fourier' '$  Inversion  Formulae  207 

the  point  x  the  integral  is  found  to  be  equal  to  the  mean  of  these  values 
instead  of  one  of  them.  Thus  in  the  last  pair  of  formulae  we  can  have 

F(x)=l        x«f>,    H(m)=  2-[l-cosw], 

77 
=  0  X>  (f>, 

but  the  integral  gives  F  (1)  =  \ . 


EXAMPLE 

If  S  (x,  t)  =  (7r/c<)~i  exp  [-  x*/4:Kt]  and  /  (x)  is  continuous  bit  by  bit  a  solution  of 
%r-  =  K  x-2,  which  satisfies  the  condition  y  =  f  (x)  when  <  =  0  and  —  oo  <  jc  <  oo ,  is 
given  by  the  formula 

v  =  4  T     flf  (x  -  *0,  0/(s0)  ^o  +  4  3  [/  (*n)  -  f(xn)] 

J   —  oo  n=l 

x  /J  [S  (a?  -a;n  -£,«)-  fl  (*  -  s,,  +  £,  *)]  <fc 

where  2/  (#„)  =  /  (zn  +  0)  +  /  (xn  —  0)  and  the  summation  extends  over  all  the  points  of 
discontinuity  of/  (x). 

§  3-12.  A  study  of  Fourier's  inversion  formula.  The  first  step  is  to 
establish  the  Biemann-Lebesgue  lemmas*. 

Let  g  (x)  be  integrable  in  the  Riemann  sense  in  the  interval  a  <  x  <  b 
and  when  the  integral  is  improper  let  |  g  (x)  \  be  integrable.  We  shall 
prove  that  in  these  circumstances 

f  b 

lim       sin  (kx) .  g  (x)  dx  =  0. 

k  ->w  J  a 

Let  us  first  consider  the  case  when  g  (x)  is  bounded  in  the  range  (a,  6) 
and  G  is  the  upper  bound  of  \g  (x)\.  We  divide  the  range  (a,  6)  into  n 
parts  by  the  points  #15  x2,  ...  xn_!  and  form  the  sums 

Sn  =  U,  (x,  -  a)  +  U2  (x2  -  x,)  +  ...  Un  (b  -  xn^)9 

sn  —  L±  (Xi  ~  a)  4-  L2  (x2  —  x^  -f  ...  Ln  (b  —  xn_j), 

where  C7r,  Lr  are  the  bounds  of  g  (x)  in  the  interval  serHl  <  x  <  xr,  so  that 
in  this  interval 

Since  &  (x)  is  integrable  we  may  choose  n  so  large  that  Sn  —  sn  <  c,  where 
c  is  any  small  positive  quantity  given  in  advance.   Now 

g  (x)  sin  Icxdx  =    S  gr  (xr^)    sin  kx .  dx  +  2    ojr  (x)  sin  kx '  dx 

*  The  proof  in  the  text  is  due  to  Prof.  G.  H.  Hardy  and  is  based  upon  that  in  Whittaker  and 
Watson's  Modern  Analysis. 


208 


Two-dimensional  Problems 


the  summations  on  the  right  being  from  r  =  1  to  r  =  n  and  the  integrations 
from  #r_,  to  xr.   With  the  same  convention 

r  b 


I    (b 

g  (x)  sin  kxdx 

I  Ja 


sin  kx .  dx 


+ 


<  2*167*  -f  Sn  -  sn  <  2nG/k  -f  c. 

Keeping  ?i  fixed  after  e  has  been  chosen  and  making  k  sufficiently  large 
we  can  make  the  last  expression  less  than  2e  and  so  the  theorem  follows 
for  the  case  in  which  g  (x)  is  bounded  in  (a,  b).  When  g  (x)  is  unbounded 
and  |  g  (x)  \  Integra ble  in  (a,  b)  we  may,  by  the  definition  of  the  improper 
integral,  enclose  the  points  at  which  g  (x)  is  unbounded  in  a  finite  number 
of  intervals  il9  i2,  ...  ip  such  that 

P 

S       I  g  (x)  I  dx  <  €. 

r~lJi, 

Now  let  G  denote  the  upper  bound  of  g  (x)  for  values  of  x  outside  the 
intervals  ir  and  let  e1?  <?2,  ...  ep+l  denote  the  portions  of  the  interval  (a,  b) 
which  do  not  belong  to  ilt  i2>  ...  iv,  then  we  may  prove  as  before  that 


g  (x)  sin  kx  .  dx 


g  (x)  sin  kx  .  dx 


P     r     I 

-f-   2        \  g  (x)  sin  kx 


dx 


<  2nG/k  4-  2e. 

Now  the  choice  of  e  fixes  n  and  6r,  consequently  the  last  expression 
may  be  made  less  than  3e  by  taking  a  sufficiently  large  value  of  k.  Hence 
the  result  follows  also  when  g  (x)  is  unbounded,  but  subject  to  the  above 
restriction. 

Some  restriction  of  this  type  is  necessary  because  in  the  case*  when  g  (x) 
is  the  unbounded  function  x~l  (1  —  x2)"*  for  which  |  g  (x)  \  is  notintegrable 
in  the  range  (—  1,  1)  we  have 

j-l  fA: 

sin  kx  g  (x)  dx  '=  TT     J0  (r)  rfr, 
J  -  1  Jo 

r  A  r  co 

and  as  k  ->  oo  J0  (r)  dr  =       JQ  (r)  dr  =  1. 

Jo  Jo 

The  next  step  is  to  show  that  if  x  is  an  internal  point  of  the  interval 
(—a,  /?),  where  a  and  /J  are  positive,  and  if  /  (x)  satisfies  in  (—a,  j8)  the 
following  conditions  : 

(1)  f  (x)  is  continuous  except  at  a  finite  number  of  points  of  dis- 
continuity, and  if  /  (x)  has  an  improper  integral  |  /  (x)  \  is  integrable  ; 

(2)  /  (x)  is  of  bounded  variation,  then 


limf 


1C  -^oo  J  -  a 

Let  us  write 


sin  k  (t  —  x) 


[x  -  0)]  =  ^ 

-  L +i: 


say. 


Fourier's  Integrals  209 

and  transform  the  integrals  by  the  substitutions  t  =  x  —  u  and  t  =  x  -f-  u 
respectively,  then 

r/3 


r  55*( 

J  —  a  &  — 

fa 

Jo 


-f- 


f  0  —  J?  ai 
^ 

Jo 


fa-t-jr 

_^)_/(z_0)]c^4-/(z-  0)          sin  ku.  du/u 

Jo 

r&-x 

«  +  ^)  -/(*  +  Q)]du+f(x+  0)          sinfai.dWtt 

Jo 


Now  let  c  denote  one  of  the  two  positive  quantities  a  -f  x,  j3  —  x,  then 

fc  r  l:c  n 

sin  ku  .  du/u  =       sin  v  .  dv/v  ->  ^  as  fc  ->  oo. 
Jo  Jo  * 

Also,  let  F  (u)  denote  one  of  the  two  functions  /  (x  —  u)  -  /  (x  —  0), 
/  (x  +  w)  —  /  (x  -j-  0),  then  jP  (0)  =  0  and  .F  (u)  is  of  bounded  variation  in 
the  interval  (0  <  u  <  c).  We  may  therefore  write 

F  (u)  =  HI  (u)  -  //2  (u), 
where  Hl  (u)  and  H2  (u)  are  positive  increasing  functions  such  that 

H,  (0)  =  a,  (0)  =  o. 

Given  any  small  positive  quantity  6  we  can  now  choose  a  positive 
number  z  such  that 

0  <  Hl  (u)  <  e,     0  <  #2  (u)  <  e, 

% 

whenever  0  <  u  <  z.    We  next  write 

sin  ku  .  F  (u)  du/u  =       sin  ku  .  F  (u)  du/u 

Jo  J  z 

-f      sin  ku  .  H  \  (u)  du/u  —       sin  ku  .  H2  (u)  du/u. 

Jo  Jo 

Let  H  (ut)  denote  either  of  the  two  functions  Hl  (u),  H2  (u)]  since  this 
function  is  a  positive  increasing  function  the  second  mean  value  theorem 
for  integrals  may  be  applied  and  this  tells  us  that  there  is  a  number  v 
between  0  and  z  for  which 

sin  ku  .  H  (u)  du/u   =    H  (z)  \    sin  ku  .  du/u 

Jo  Jv 

I    (kz 

—  H  (z)          sin  s  .  dsls  . 

I  Jkv 

roo  r  oo 

Since       sin  s  .  dsjs  is  a  convergent  integral,       sin  s  .  dsjs  has  an  upper 

JO  JT 

bound  B  which  is  independent  of  r  and  it  is  then  clear  that 


11; 


sin  ku  .  H  (u)  du/u 


<  2BH  (z)  <  2B€. 


210  Two-dimensional  Problems 

By  the  first  lemma  k  may  be  chosen  so  large  that 

sin  ku  .  F  (u)  du/u   <  e, 
and  so  we  have  the  result  that 

re 

lim      sin  ku  .  F  (u)  du/u  =  0. 

Ar-voo  Jo 

It  now  follows  that 

k  -><X>  J  -a  *          X  4 

To  extend  this  result  to  the  case  in  which  the  limits  are  —  oo  and  oo 
we  shall  assume  that  for  x  >  j3 


where  P1  (x)  and  P2  (x)  are  positive  functions  which  decrease  steadily  to 
zero  as  x  increases  to  oo.  A  similar  supposition  will  be  made  for  the  range 
x  <  —  a,  the  positive  functions  now  being  such  that  they  decrease  steadily 
to  zero  as  x  decreases  to  —  oo.  Since 


Pi(t) 
t-  x' 


(x  <  /?  <  t  <  y) 


is  a  positive  decreasing  function  of  t  for  t>  ft  we  may  apply  the  second 
mean  value  theorem  for  integrals  and  this  tells  us  that 

x)         ^pl(p)  {*sink(t_x)dt  +  P> 

- 


Now  let  \Pl(x)\<  M  for  x  >  /?,  then 

M 


D  m   ,, 

Pl(t)dt 


J-TQ  -  , 
k(p-x) 


-f 


sin 


By  making  k  large  enough  we  can  make  4M/k  (j8  —  x)  as  small  as  we 
please ;  moreover,  this  quantity  is  independent  of  y,  and  so  we  can  conclude 

that  ("sink  (t-x) 

lim   I         —  _  Px  (t)  dt  —  0. 

k  ->oo  J  P  t        •£ 

Similar  reasoning  may  be  applied  to  the  integral  involving  P2  (t)  and 
to  the  integrals  arising  from  the  range  t  <  —  a.  It  finally  follows  that 


foo 

im 

_>00  J-C 


lim 


or 


sin  k  (t  —  x) 

x,       t  —  x 

}oo         rk 
dt     ' 
-oo     JO 


~  x)f(t)ds. 


Fourier's  Integrals  211 

To  justify  a  change  in  the  order  of  integration  it  will  be  sufficient  to 
justify  the  change  in  the  order  of  integration  in  the  repeated  integral 


I  °°  dt  f  cos  s  (t  -  x)  Pl  (t)  ds, 

Jq         JO 


where  q  >  /?,  for  the  other  integral  with  limit  —  oo  may  be  treated  in  the 
same  way  and  a  change  in  the  order  of  integration  for  the  remaining 
integral  between  finite  limits  may  be  justified  by  the  standard  analysis. 
Now  let  us  assume  that 

\mPi(t)dt  ......  (A) 

Jq 

exists,  then 
f  "dt  j  k  cos  s(t-  x)  Pl  (t)  ds~  \K  ds  Tcos  s(t-x)  Pl  (t)  dt    <  2k  f  "X  (t)  dt. 

Jq        JO  JO          Jq  Jq 

But,  since  the  integral  (A)  exists  we  can  choose  q  so  large  that 


is  as  small  as  we  please.  The  order  of  integration  can  therefore  be  changed 
and  so  we  have  finally 


TT/(X)  =  \ds\     coss(t~  x)f(t)  dt. 

JO  J  -oo 


The  assumptions  which  have  been  made  are: 

(1)  For   x  >  p,  f(x)  =  Pj  (x)  -  P2  (x),    where   P1  (x)    and   P2  (x)    are 
positive  decreasing  functions  integrable  in  the  range  ()8,  oo). 

(2)  A  corresponding  supposition  for  x  <  —  a. 

(3)  /  (x)  of  bounded  variation  in  a  range  enclosing  the  point  x. 

(4)  /  (x)  discontinuous  at  only  a  finite  number  of  points  in  (—  a,  f3) 
and  j  /  (x)  |  integrable  in  (—  a,  /2). 

§  3-13.  To  illustrate  the  method  of  summation  we  shall  try  to  find  a 
potential  which  is  zero  when  x  =  0  and  when  x  =  1 .  We  shall  be  interested 
here  in  the  case  when  the  potential  has  a  logarithmic  singularity  at  the 
point  x  =  x',  y  =  0. 

We  first  note  that  M  (m,  x')  is  a  simple  combination  of  primary 
solutions  and  by  an  extension  of  the  method  of  images  used  in  the  solution 
of  physical  problems  by  means  of  primary  solutions  we  may  satisfy  the 
boundary  condition  at  x  =  0  by  means  of  a  simple  potential  of  type 
M  (m,  x')  —  M  (m,  —  x').  This  can  be  written  in  the  form 

2e~my  sin  (mx)  .  sin  (mx')> 

and  it  is  readily  seen  that  the  boundary  condition  at  x  =  1  may  be  satisfied 
by  writing  m  =  mr,  where  n  is  an  integer.  We  now  multiply  by  a  function 

14-2 


212  Two-dimensional  Problems 

of  n  and  sum  over  integral  values  of  n.  To  obtain  a  series  which  can  be 
summed  by  means  of  logarithms  we  choose/  (n)  =  l/n  so  that  our  series  is 

co       1 

K  =   £    -  e~nny  [cos  /ITT  (x  —  x')  —  cos  n-n  (x  -f  #')]  • 
n~-in 

If  //  >  0  the  sum  of  this  scries  is* 

p       i  ,      cosh  (rry)  —  cos  TT  (#  4-  a;7) 
~  COsh  (TT?/)  —  COS  TT  (x  ~  x')  ' 

To  extend  our  solution  to  negative  values  of  y  we  write  it  in  the  form 

on        O 

V  —    £        g~w;r|y|  sjn  {n7lX}  sin 


The  expression  for  V  may  be  written  in  an  alternative  form 


which  shows  that  it  may  be  derived  from  two  infinite  sets  of  line  charges 
arranged  at  regular  intervals. 

This  expression  shows  also  that  the  potential  V  becomes  infinite  like 
—  \  log  [(x  —  x')2  -f  i/'2]  in  the  neighbourhood  of  x  =  x'  ',  y  —  0,  it  thus 
possesses  the  type  of  singularity  characteristic  of  a  Green's  function  and 
so  we  may  adopt  the  following  expression  for  the  Green's  function  for  the 
region  between  the  lines  x  —  0,  x  —  1,  when  the  function  is  to  be  zero  on 

these  lines  oo    j 

G  (x,x'\y,y')  =  £  -  e  -nrr\v-v'\  s[n  (nnx)  sin 

n  -  1  n 

A  corresponding  solution  of  the  equation 


is  obtained  by  writing 

exp  [—  |  y  —  y'  \  (nV2 
in  place  of  exp  [—  UTT  \  y  —  y'  \] 

and  2n/(n*  -  k*/7r2) 

in  place  of  the  factor  2/n. 

§  3-14.  As  another  illustration  of  the  use  of  the  simple  solutions  of 
Laplace's  equation  we  shall  consider  the  problem  of  the  cooling  of  the  fins 
of  an  air-cooled  airplane  engine  when  the  fins  are  of  the  longitudinal  type. 

The  problem  will  be  treated  for  simplicity  as  two-dimensional. 

A  fin  will  be  regarded  as  rectangular  in  section,  of  thickness  2r,  and  of 
length  a.  Assuming  that  the  end  x  =  0  is  maintained  at  temperature  00  by 
the  cylinder  of  the  engine  and  that  it  is  sufficient  to  assume  a  steady  state, 

329      o26 
the  problem  is  to  find  a  solution  of  ^~2  -f  ^  =  0  and  the  boundary  con- 

*  See,  for  instance,  T.  Boggio,  Rend,  '  Lombardo  (2)  42:611-624  (1909). 


Cooling  of  Fins  213 

ditions  *  --  =  0  along  y  =  0,  k  ~—  =  —  q0  along  y  =  T,  A  y-  =  —  <?#  along 
#  =  a. 

The  first  two  of  these  three  conditions  are  satisfied  by  writing 

00 

0=2  Am  cosh  [sm(x-cm)]  cos  (smy),     ksm r  tan  (sm r)  -  q. 

m-l 

This  equation  gives  oo1  values  of  sm  and  when  sm  has  been  chosen  the 
corresponding  value  of  cm  is  given  uniquely  by  the  equation 

ksm  tanh  [sm  (cm  -  a)]  =  y 
which  will  ensure  that  the  third  condition  is  satisfied. 

To  make  0  =  0Q  when  x  =  0  we  have  finally  to  determine  the  constant 
coefficients  Am  in  such  a  way  that 

00 

00  =   S  ^4W  cosh  (smcm)  cos  (sm?/). 
/«  - 1      • 

This  may  be  done  with  th£  aid  of  the  orthogonal  relations 
I   cos  (ysm)  cos  (ysn)  dy  =  0,     m  ^  n 


m  =  n. 


Therefore  Am  =  400  sech 


cosh  (^y)  sin  (smr). 

v  inj)        Vm   ; 


Harper  and  Brown  derive  from  this  expression  a  formula  for  the 
effectiveness  of  the  fin,  which  they  define  as  the  ratio  H/HQ)  where 

Hn  =  2q  (a  +  r)  0n,     H  =  a  \0dS. 


For  numerical  computations  it  is  convenient  to  adopt  an  approximate 
method  in  which  the  variation  of  9  in  the  y  direction  is  not  taken  into 
consideration.  Results  can  then  be  obtained  for  a  tapered  fin. 

The  approximate  method  has  been  used  by  Binnie|  in  his  discussion 
of  the  problem  for  the  fins  of  annular  shape  which  run  round  a  cylinder 
barrel. 

§  3-15.  For  some  purposes  it  is  useful  to  consider  simple  solutions  of 
a  complex  type.  Thus  the  equation 

dv  _     d2v 
Si  =  "  dx* 

*  The  formal  solution  is  obtained  by  D.  R.  Harper  and  W.  B.  Brown  (N.A.C.A.  Report, 
No.  158,  Washington,  1923),  but  is  not  used  in  their  computations, 
f  Phil.  Mag.  (7),  vol.  n,  p.  449  (1926). 


214  Two-dimensional  Problems 

is  satisfied  by  v  =  Ae"**(l+l>**, 

if  2i//?2  =  cr.    Retaining  only  the  real  part  we  have* 

v  =  Ae-*xco&  (at  -  px).  ......  (A) 

This  solution  is  readily  interpreted  by  considering  a  viscous  liquid  which 
is  set  in  motion  by  the  periodic  motion  of  the  plane  x  =  0,  the  quantity 
v  being  velocity  in  one  direction  parallel  to  this  plane  (§  2-56).  The  pre- 
scribed motion  of  the  plane  x  =  0  is 

v  =  A  cos  at  =  F,  say. 

The  vibrations  are  propagated  with  velocity  or/j3  in  the  direction  per- 
pendicular to  the  plane  but  are  rapidly  damped  ,  for  the  amplitude  diminishes 
in  the  ratio'  e~2rr  as  the  wave  travels  a  distance  of  one  wave-length  27r/j3. 
For  an  assigned  value  of  a  this  wave-length  is  very  small  when  v  is  very 
small,  when  v  is  assigned  the  wave-length  is  very  small  if  a  is  very  large. 

The  equation  (A)  has  b,een  used  by  G.  I.  Taylorf-to  represent  the  range 
of  potential  temperature  at  a  height  x  in  the  atmosphere,  the  potential 
temperature  being  defined  as  usual,  as  the  temperature  which  a  mass  of 
air  would  have  if  it  were  brought  isentropically  (i.e.  without  gain  or  loss 
of  heat  and  in  a  reversible  manner)  to  a  standard  pressure. 

The  following  examples  to  illustrate  the  use  of  the  solution  (A)  are 
given  by  G.  Greenf. 

Suppose  that  two  different  media  are  in  contact,  the  boundary  surface 
being  x  -=  a  and  the  boundary  conditions 

v    9^,  „    3V2          x 

v>=""  **£  =  *•&  for*=a- 

Let  there  be  a  periodic  source  of  "plane  -waves"  on  the  side  x,  then 
the  solution  is  of  type 

Vl  =  6e~^x  cos  (at  -  fax)  -f  AOeW*-**)  cos  [at  +  fa(x—  2a)],     x  <  a, 

v2  -  Bde-t*(*-c)  cos  [at  -  fa  (x  -  c)],       x  >  a, 
where  c  =  a[l  -  (i/^)*],     fa  =  (cr/2^)*,     &  =  (<7/2i/8)*, 

pA  =  K,  vS  -  #2  vX>      PB  =  2^  yV2>      P  =  #1  V"2  +  #2  vX- 
There  is,  of  course,  the  physical  difficulty  that  the  expression  for  the 
incident  waves  becomes  infinite  when  x  =  —  oo. 

If  we  take  the  associated  problem  in  which  the  incident  waves  corre- 
spond to  a  periodic  supply  of  heat  q  cos  at  at  the  origin,  the  solution  is 


(y2/a)*  Be-**(x-c)  cos  (kt  -  fax  +  fac  -  7r/4), 
where  A  and  B  have  the  same  values  as  before.   It  is  noteworthy  that  A 

*  The  theory  is  due  to  Stokes,  Papers,  vol.  m,  p.  1.  See  Lamb's  Hydrodynamics,  p.  586. 
t  Proc.  Eoy.  Soc.  London,  A,  vol.  xciv,  p.  137  (1918). 
t  G.  Green,  Phil  Mag.  (7),  vol.  in,  p.  784  (1927). 


Fluctuating  Temperatures  215 

and  B  are  independent  of  a  and  that  when  the  expressions  in  these  solutions 
are  integrated  with  respect  to  a  from  0  to  oo  the  physically  correct  solution 
for  the  case  of  the  instantaneous  generation  of  a  quantity  of  heat  q  at  the 
origin  at  time  t  ==  0  is  obtained  in  the  form 


EXAMPLES 


1.  In  the  problem  of  the  oscillating  plane  the  viscous  drag  exerted  by  the  fluid  is,  per 
unit  area, 

.    /  1   /717\ 

(sin  at  —  cos  at). 

[Rayleigh.] 

2.  Discuss  the  equations 


(  V  -H  --  ~5T  \ 


a)  =*  a  sin  <^,     (<^  a  constant), 

where  Q  is  a  constant  representing  the  angular  velocity  of  the  earth,  and  <f>  is  the  latitude. 

[V.  W.  Ekman.] 

§  3-16.   The  solution  (A  of  §  3-15)  may  be  generalised  by  regarding  A 
as  a  function  of  /?  and  then  integrating  with  respect  to  /S. 
A  solution  of  a  very  general  character  is  thus  given  by 

v  =  I  °V*te  cos  [3x  - 


+        e  -^  sin  [jSa;  -  2j/j32*]  e/r  (j8)  rfjS, 

Jo 

where  (/>  (j3)  and  ^  (j8)  are  arbitrary  functions  of  a  suitable  character. 
Solutions  of  this  type  have  been  used  by  Rayleigh  and  by  G.  Green. 

Some  useful  identities  may  be  obtained  by  comparing  solutions  of 
problems  in  the  conduction  of  heat  that  are  obtained  by  two  different 
methods  when  the  solution  is  known  to  be  unique. 

For  instance,  if  we  use  the  method  of  simple  solutions  we  can  construct 
a  solution  2jr 

t;  =  -     S          et{*-«n-^a/(f)df 

7Tn»-oo   Jo 

which  is  periodic  in  x  with  the  period  2?!. 

When  t  =  0  the  series  is  simply  the  Fourier  series  of  the  function  /  (x) 
and  the  inference  is  that  with  a  suitable  type  of  function  /  (x)  our  solution 


216  Two-dimensional  Problems 

is  one  which  satisfies  the  initial  condition  v  =  /  (x)  when  t  —  0.  Now  such 
a  solution  can  also  be  expressed  by  means  of  Laplace's  integral 


t;  - 


rco 

e      M    f(t)d(9 

J  -co 

and  this  may  be  written  in  the  form 


00  ft,  --- 

2  e          4*< 

JO 


71=  —  oo 


When  the  order  of  integration  and  summation  can  be  changed,  a  comparison 
of  the  two  solutions  indicates  that  , 

(x  -  £_+  gftTT,2 

2    e"1**-^-"21'*  =  (77/1/0*    S 


This  identity,  which  is  due  to  Poisson,  has  recently,  in  tho  hands  of 
Ewald,  become  of  great  importance  in  the  mathematical  theory  of  electro- 
magnetic waves  in  crystals.  The  identity  can  be  established  rigorously  in 
several  ways  : 

(1)  With  the  aid  of  Fourier  series. 

(2)  By  the  calculus  of  residues. 

(3)  By  the  theory  of  elliptic  functions  (theta  functions). 

(4)  By  means  of  the  functional  relation  for  the  f  -function,  Riemann's 
method  of  deriving  this  functional  relation  being  performed  backwards. 

An  elementary  proof  based  on  the  equations 

x 

n 


>ex     astt->oo     if  #„->#,  ......  (1) 

2~2nnH  j^^-ig-acz    as  n  -^  oo    if  rn~^  ->  x    ......  (2) 

\n  -h  r)  v  ; 

has  been  given  recently  by  Polya*. 

We  have2n~1<  n\  for  n=  1,2,3,...  and  so,  forO<.r<  1, 

e**  =  1  +  ~  +  ...  <  1  +  2x  +  2x*  +  ...  =  1±5. 
1  !  1  —  x 

Also,  for  0<  x<  {, 

1  4-  T  Sr3  7^3 

^J_  =  1  +  2x  +  2x2  +  ~  —  <  1  +  2x  4-  2x2  +   ^ 


Therefore  e-2x-x*  <     _~-    <  e-2* 

*  G.  P61ya,  Berlin,  Akad.  Wiss.  Ber.  p.  158  (1927). 


Frisson's  Identity  217 

On  account  of  the  symmetry  of  the  binomial  coefficients  it  is  sufficient  to 
prove  (2)  for  r  >  0.  In  this  case  .        j.    ,        2\        /       r  —  1\ 


.        j.    ,        2\        /       r  — 
n      (l  ~  n)  V  ~  n)  '"  \l  ~  ~^ 


r/        1W        2X       /        r-_l\ 

V        n)\       n)       \  n    / 


V' 

the  upper  estimate  in  (B)  having  been  applied.  A  use  of  the  lower  estimate 
gives  an  analogous  result,  which,  on  account  of  the  fact  that  r4n~3  ->  0,  com- 
pletes the  proof  of  (2). 

Putting  x  =  ZCDV  =  ze2irtv'1  in  the  identity 


k         /      fyvn      \ 

we  obtain  S      [(V(za>v)  4-  1  A/(z"")]2m  =  1    %    (         ,   )  zl¥, 

-1<2»<1  v--k  \m  +  W/ 

where  k  =  [m/l]  is  the  integral  part  of  m/l. 

Now  let  s  be  an  arbitrary  fixed  complex  number  and  t  a  fixed  real  positive 
number.  Putting  I  =  V[(mt)]>  z  =  gS^»  an(l  dividing  the  series  by  22m,  we  obtain 
the  relation 


8P~\ 
2      coshM--~^n  =       2       Jl  +  rA™l^+      I 

[Vim] 


^ (C) 


Applying  the  limit  (1)  on  the  left  and  (2)  on  the  right  we  finally  obtain 

a  +  2viv 

00  -jj-y CO 

V— -00  —00 

which  is  a  form  of  Poisson's  formula. 

To  justify  the  limiting  process  which  has  just  been  performed  in  which  the 
limit  is  taken  for  each  term  separately,  it  is  sufficient  to  find  a  quantity  inde- 
pendent of  m  which  dominates  each  term  in  each  series. 

There  is  little  difficulty  in  finding  a  suitable  dominating  quantity  for  the 
terms  on  the  right-hand  side,  but  to  find  a  suitable  quantity  for  the  terms  on 
the  left-hand  side  x>f  the  equation  P61ya  finds  it  necessary  to  prove  the  following 
lemma.  Given  two  constants  a  and  b  for  which  a  >  0,  0  <  b  <  IT,  we  can  find 
two  other  constants  A  and  B  such  that  A  >  0,  B  >  0  and 

|  cosh  z  |  <  eAxZ~Bv*9 
when  —  a <  x <  a,  —  b<y<b  and  z  =  x  +  iy.   We  have,  in  fact, 

|  cosh  z  |2  =  \  (1  +  cosh  2x)  -  sin2*/, 
but,  for  —  a  <  x  <  a,  we  have 

i  <1  4-  cosh  2x)  <  1  -f-  i  2  — T^-T-r~  =  1  -f  2Ax2y  say. 
n-i     (2n)\ 


218  Two-dimensional  Problems 

On  the  other  hand,  since  sin  y/y  decreases  as  y  increases  from  0  to  TT,  we  have 
for  -  6  <  y  <  6, 

^  >  ™*  =  Vfff,  8ay,   V25>0. 

It  follows  from  the  inequalities  that  have  just  been  established  that 
|  cosh  z  |2<  1  +  24z2  -  2£?/2<  e2<^2-*"2), 

and  this  proves  the  lemma. 

To  apply  the  lemma  to  the  series  (C)  we  note  that  in  the  first  member 

|   TTV/l  |  <   77/2, 

we  therefore  take  b  =  77/2. 

If  s  is  real,  the  sum  in  the  first  member  of  (C)  is  dominated  by  the  series 


I    e-gr         F-. 

V—  —00 

It  is  easy  to  dominate  this  series  by  one  free  from  m.  The  case  in  which 

s  is  not  real  can  also  be  treated  in  a  similar  manner. 

\ 

EXAMPLES 

1.  If  P  =  [  °°  e~x*  [cos  (xO)  -  sin  (x9)]  cos  (2**02)  d6, 

JO 

Q  =  /  °°  e-^  [cos  (x^)  +  sin  (xd)]  sin  (2K^2)  d0, 

JO 
show  by  partial  integration  that 

xP=>-2KtlQ,    xQ  =  -1Kt*£. 
ox  ox 

Show  also  that,  as  t  ->  0, 

4P-v4Q->rr*(^ri, 
and  that  consequently 

4P  =  4C  =  7r*(,cO~>e-a;2/4**. 

2.  If  C  -  (  °°    e-^v  cos  (y2  -  a2)  dt/, 

7  -« 

/•  oo 

^f^          e-^  sin  (y2  -  a2)  (^y, 
J  -a 

prove  that  (7  +  S  =  V(w/2),     0-^  =  2  I  *  e29*  dd.  [G.  Green.] 

§  3-17.  Conduction  of  heat  in  a  moving  medium.  When  the  temperature 
depends  on  only  one  co-ordinate,  the  height  above  a  fixed  horizontal  plane, 
and  the  vertical  velocity  of  the  medium  is  w,  the  equation  of  conduction  is 

do      90_   d*e 

dt  +  Vdy~K~dy*'  ......  (A) 

where  K  is  the  diffusivity.    When  v  is  constant  the  equation  possesses  a 
simple  solution  of  type 

/i  +  v\  =  *A2, 


Conduction  in  a  Moving  Medium  219 

which  may  be  generalised  by  summation  or  integration  over  a  suitable  set 
of  values  of  /*.  In  particular,  if  we  regard  0  as  made  up  of  periodic  terms 
and  generalise  by  integration  over  all  possible  periods,  we  obtain  a  solution 

roo 

0  =,       eay  [f(a)  sin  (by  -f-  at)  +  g  (a)  cos  (by  +  at)]  da, 

.70" 

where  2*a  =  v  —  w,      bw  =  —  a, 


and  /  (a),  g  (a)  are  &c*itable  arbitrary  functions.  The  integral  may  be  used" 
in  the  Stieltjes  sense  so  that  it  can  include  the  sum  of  a  number  of  terms 
corresponding  to  discrete  values  of  a. 

When  v  varies  periodically  in  such  a  way  that  v  =  u  (1  4-  r  COB  at), 
where  u,  r  and  a  are  constants,  a  particular  solution  may  be  obtained  by 


......  (B) 

where  /  (t)  is  a  function  which  is  easily  determined  with  the  aid  of  the 
differential  equation.  When  v  is  an  arbitrary  function  of  t  the  equation 
(A)  has  a  simple  solution  of  type 


which  may  be  generalized  into 

0  =  [  °°    ewl»  -'""I  ~  •**  F(s)ds 

t  J  —  oo 

where  F(s)  is  a  suitable  arbitrary  function  of  s. 

The-solution  (B)  has  been  used  by  McEwen*  for  a  comparison  of  the 
results  computed  from  theory  with  the  results  of  a  series  of  temperature 
observations  made  off  Coronado  Island  about  20  miles  from  San  Diego  in 
California.  The  coefficient  K  is  to  be  interpreted  as  an  "eddy  conductivity" 
in  the  sense  in  which  this  term  is  used  by  G.  I.  Taylor.  This  is  explained 
by  McEwen  as  follows  : 

At  a  depth  exceeding  40  metres  the  direct  heating  of  sea  water  by  the 
absorption  of  solar  radiation  is  less  than  1  per  cent,  of  that  at  the  surface. 
Also,  the  temperature  range  at  that  depth  would  bear  the  same  proportion 
to  that  at  the  surface  if  the  variation  in  rate  of  gain  of  heat  were  due  only 
to  the  variation  in  this  rate  of  absorption.  The  direct  absorption  of  solar 
radiation  cannot  then  be  the  cause  of  the  observed  seasonal  variation  of 
temperature,  which  amounts  to  5°  C.  at  a  depth  of  40  metres  and  exceeds 
1°  at  a  depth  of  100  metres.  Laboratory  experiments  show,  moreover, 

*  Ocean  Temperatures,  their  relation  to  solar  radiation  and  oceanic  circulation  (University  of 
California  Semicentennial  Publications,  1919). 


220  Two-dimensional  Problems 

that  the  ordinary  process  of  heat  conduction  in  still  water  is  wholly  in- 
adequate to  produce  a  transfer  of  heat  with  sufficient  rapidity  to  account 
for  the  whole  phenomenon.  It  is  now  generally  recognised  that  a  much 
more  rapid  transfer  of  heat  results  from  an  alternating  vertical  circulation 
of  the  water  in  which,  at  any  given  instant,  certain  portions  of  the  water 
are  moving  upward  while  others  are  moving  downward.  The  resultant 
flow  of  a  given  column  of  water  may  be  either  upward  or  downward,  or 
may  be  zero.  The  motion  may  be  described  as  turbulent  and  a  vivid  picture 
of  the  process  may  be  obtained  by  supposing  that  heat  is  conveyed  from 
one  layer  to  another  by  means  of  eddies.  This  complicated  process  produces 
a  transfer  of  heat  from  level  to  level  which,  when  analysed  statistically, 
will  be  assumed  to  be  governed  by  the  same  law  as  conduction  except  that 
the  "eddy  conductivity"  or  "JMischungsintensitat  "  will  depend  mainly 
on  the  intensity  of  the  circulation  or  mixing  process. 

An  equation  which  is  more  general  than  (A)  has  been  obtained  by 
S.  P.  Owen*  in  a  study  of  the  distribution  of  temperature  in  a  column  of 
liquid  flowing  through  a  tube. 

Assuming,  as  an  inference  from  Nettleton's  experiments,  that  the  shape 
of  the  isothermals  is  independent  of  the  character  of  the  flow,  Owen  con- 
siders an  element  of  length  8?y  fixed  in  space  and  estimates  the  amounts 
of  heat  entering  and  leaving  the  element  across  its  two  faces  perpendicular 
to  the  y-axis  to  be 

*{-*%+'»'* 
and          ^{-*^(0  +  ^ 

respectively,  where  A  is  the  area,  p  the  perimeter  of  the  cross-section  of 
the  tube,  9  the  temperature  of  the  element,  E  the  emissivity,  k,  p  and  s 
the  thermal  conductivity,  density,  and  specific  heat  of  the  liquid  respec- 
tively, and  where  00  is  the  temperature  of  the  enclosure  which  surrounds 
the  tube. 

Owen  thus  obtains  the  equation 


A   k     *  ~  psv  y  8y  -EP(0-  *o)  %  =  Aps     Sy, 

«2n-»f-?('-*o)=4' 

dy*        dy     Aps{         °'      dt* 
where  a2  -  k/ps. 

EXAMPLES 

1.   Prove  that  a  temperature  0  which  satisfies  the  equation 

de      de      dze 

Ht  ri~  =  K  3^*  ' 

and  the  conditions  ^ 

0  =  0  when  z,  =  0,     0  «  0,  when  y  *  6,     0  .  0  when  f  -  0, 
*  Proc.  London  Math.  Soc.  vol.  xxm,  p.  238  (1925). 


Wave  Propagation  by  an  Electric  Cable  221 

is  given  by  the  formula 


ttnz*z*!b*)  +  (i?«/4«)}M. 
[Somers,  Proc.  P%$.  &>c.  London,  vol.  xxv,  p.  74  (1912);  Owen,  foe.  ci7.] 


2.  If  in  the  last  example  the  receiver  is  maintained  at  a  temperature  which  is  a  periodic 
function  of  the  time,  so  that  the  condition  9  =  &l  when  y  —  b  is  replaced  by  d  =  6  cos  a>t 
when  y  =  b,  the  solution  is 

B  =  aev(i/-&)/2*  (cosh  2nb  —  cos  2w6)-1  [(cos  m£  cosh  wiy  —  cos  my  cosh  TI£)  cos  o>£ 
—  (sin  wi|  sinh  7117  —  sin  m^  sinh  T?£)  sin  a>t] 

4-  2e&~2    2     ( 

p-1 
where 


=  {(v/2K)*  -f  (a;/^)2}         {i  tan-1  (4^a./?;2)}. 
7i  sin 

§  3-18.  Theory  of  the  unloaded  cable.  Consider  a  cable  in  the  form  of  a 
loop  (Fig.  13)  having  an  alternator  A  at  the  sending  end  and  a  receiving 
instrument  B  at  the  receiving  end. 
We  shall  suppose  that  the  alter 
nator  is  generating  a  simple  periodic 
electromotive  force  which  may  be 
represented  as  the  real  part  of  the  ° 

expression  Eelni,  where  E  and  n  are  constants.  Naturally,  we  arc  interested 
only  in  the  real  part  of  any  complex  quantity  which  is  used  to  represent 
a  physical  entity. 

Now,  if  CSx  is  the  capacity  of  an  element  of  length  8x  with  regard  to 
the  earth,  the  capacity  of  a  length  ox  with  regard  to  a  similar  element  in 
the  return  cable  must  be  ^Cox.  Hence,  if  70  is  the  current  in  the  alternator 
and  VQ  the  potential  difference  of  the  two  sides  of  the  cable  at  the  sending 

end'  lcaF0       a/0 

~*C  W~~~  ~dx' 

Now  VQ  is  the  difference  between  the  generated  electromotive  force 
Eeint  and  the  drop  in  voltage  down  the  alternator  circuit  and  a  capacity 
CQ  in  series  with  it,  consequently  we  have  the  equation 


+  F0  = 


Assuming  that  /  can  be  expressed  as  the  real  part  of  X  (x)  eint  and 
that  /  =  /0  at  the  receiving  end,  we  find  on  differentiating  the  last  equation 
with  respect  to  t  and  multiplying  by  C0  , 

(1  -  CQL0n*  +  inCQRQ)  70  -f  C0      -° 


222  Two-dimensional  Problems 

Hence  the  boundary  condition  at  the  sending  end  is 


- 

ox 

where  hQC0  =  0(1-  C0L0n*  -f  inC0R0). 

Similarly,  if  /j  is  the  current  at  the  receiving  end  and  if  the  receiving 
apparatus  is  equivalent  to  an  inductive  resistance  (L19  RJ  in  series  with  a 
capacity  Cl  ,  we  have  the  boundary  condition 

9/1- 


where  A^  =  C  (1  -  C^n2  + 

Assuming  that  there  is  no  leakage,  the  differential  equation  for  /is  * 


and  if  X  =  K1  cos  /z  (I  —  x)  +  K2  sin  /z  (I  —  x), 

where  I  is  the  distance  between  the  alternator  and  receiving  instrument, 
and  Kl9  K2  are  constants  to  be  determined,  we  have 

/x2  =  C  (n*L  -  inR). 
Writing  /x  =  a  -f  i/3,  where  a  and  /J  are  real,  we  have 

a2  -  )82  =  LCn2,     2ap  =  -  CRn, 
and  so,  if  R2  -f  n*L2  =  G2,  we  have 

2«2  =Cn(G  +  nL),       2^82  -  Cn  (G  -  nL). 
When  nL  is  large  in  comparison  with  R  we  may  write 

and  we  have 

a  = 

the  wave-  velocity  being  (CL)~^.  In  this  case  the  wave-velocity  and 
attenuation  constant  are  approximately  independent  of  the  frequency, 
consequently  a  wave-form  built  up  from  waves  of  high  frequency  travels 
with  very  little  distortion. 

The  constants  JK1  and  K2  are  easily  determined  from  the  boundary- 
conditions  and  we  find  that 


where  F  =  (/^/^  —  4/I2)  sin  pi  +  2/*  (h0  +  k^  cos  pi. 

When  E  =  0  the  differential  equation  possesses  a  finite  solution  only 
when  F  =  0  and  this,  then,  is  the  condition  for  free  oscillations.  The  roots 
of  the  equation  F  =  0,  regarded  as  an  equation  for  n,  are  generally  complex. 

*  Our  presentation  is  based  upon  that  of  J.  A.  Fleming  in  his  book,  The  propagation  of  electric 
currents  in  telephone  and  telegraph  conductors. 


Roots  of  a  Transcendental  Equation  223 

This  may  be  seen  by  considering  the  special  case  when  (70  =  Cl  =  oo.  This 
means  that  there  are  short  circuits  in  place  of  the  transmitting  and  re- 
ceiving apparatus. 

We  now  have  A0  =  Ax  =  0,  p?  sin  [d  =  0,  and  if  we  satisfy  this  equation 
by  writing  p,l  —  SIT,  where  s  is  an  integer,  the  equation 
sW  =  pip  =  l*C  (n*L  -  inR) 

gives  complex  values  for  n. 

When  J?0  =  Rl  =  R  the  roots  of  the  equation  for  n  are  all  real.  This 
may  be  proved  with  the  aid  of  the  following  theorem  due  to  Koshliako  v  *  . 

m  n 

Let  <£0  -f  iiff0  -  2  ms  log  (z  -  z8)  -  S  k8  log  (z  -  £s) 

5-1  S-l 

be  the  complex  potential  of  the  two-dimensional  flow  produced  by  a 
number  of  sources  and  sinks,  the  sources  being  all  above  the  axis  of  x  and 
the  sinks  all  on  or  below  the  axis  of  x. 

Writing  zs  =  aB  +  ibs,       £,  =  £,-  iij,  , 

where  a8  ,  b8  ,  gs  ,  r}3  are  all  real,  we  shall  suppose  that 

bs  >  0,       T]S  >  0,       ms  >  0,       ks  >  0. 
Now  suppose  that  when  x  is  real  and  complex 

n    Ir'  _  y  ^w*t 

n,  r-fe  -/<*>  +  v(»). 

S-l    I*'   —     fcj     * 

where  /  (x)  and  0  (x)  are  real  when  x  is  real.  If  we  superpose  on  the  flow 
produced  by  the  sources  and  sinks  a  rectilinear  flow  specified  by  the  stream- 
function  fa  =  x  —  y  tan  o>,  the  stream  -function  of  the  total  flow  is 
^  =  i/jQ  -f  fa  and  the  points  in  which  a  stream  -line  iff  =  6  cuts  the  axis  of 
x  are  given  by  the  transcendental  equation 


or  g  (x)  cos  (x  -  d)  +  f  (x)  sin  (x  -  0)  =  0. 

We  wish  to  show  in  the  first  place  that  the  roots  of  this  equation  are 
all  real.   Writing 


G  (x]  -  iF  (x 

IT  (X)  -  lit    (X     =  _    ,  . 

s«l  ^         bs         *Vs/ 

we  have  G  (a:)  -f-  iF  (x)  =  ez  (x-d)  [/(*?)  +  *V  («)], 

0  (a?)  -  *T  (x)  =  e'  <«-*>  [/  (x)  -  t^  (a?)], 
^  («)  =  /  (»)  sin  (a;  ^  «)  +  gr  (a:)  cos  (x  -  0), 
G(x)=f  (x)  cos  (a:  -  6)  -  g  (x)  sin  (a;  -  6). 

*  Mess,  of  Math.  vol.  LV,  p.  132  (1926).   Koshliakov  considers  only  the  case  m,  -  1,  k,  =  0 
without  any  hydrodynamical  interpretation  of  the  result. 


224  Two-dimensional  Problems 

Hence,  if  z  =  x  4-  iy  is  a  root  of  the  equation  F  (z)  —  0,  we  have  for 
this  root  % 

Now  let  M^  and  M2  be  the  moduli  of  the  -expressions  on  the  two  sides 
of  the  equation,  then  the  equation  tells  us  that  M^  =  J/22,  but 


M  *  _  e- 

1  ~e 


and  from  these  equations  it  appears  that  y  >  0,  M-f  <  M22,  while  if 
y  <  0,  M !2  >  J/22.  Hence  we  must  have  y  =  0,  and  so  the  roots  of  the 
equation  ^T  (2)  =  0  are  all  real. 

Let  us  next  determine  the  effect  on  the  roots  of  varying  the  value  of  6. 

If  a;  is  a  real  root  of  the  equation  F  (x)  =  0,  we  have 

(dx/d9)  [/'  (x)  sin  (x  -  9)  +  g'  (x)  cos  (x  -  0)  -f-  G  (x)]  -  G  (x), 

,     .  cos  (x  —  0)     sin  (x  —  6)         1 

f(*\  r^i        T^T'i' 

therefore    (<te/rf0)  [/  (a?)  y'  (x)  -  f  (x)  g  (x)  +  {(7  (a;)}2]  -  [6y  (a-)]2. 
Now  r;  ,     -!,,-=  S  (  *     .,   -   - 

i/'yi     I     Q  rt  I  'Y\  •.\/y  ATT     —   *?  /i          i* 

/    it*/ 1   "f~    t-j/  (».</  /          ^ _a  j  \»</          tt'g          ^^5        ^ 

/  (a:)  —  ig  (x)       s=i\x  ~  as  +  ^5      ^ 
therefore 

»-/»£(*)       > 


The  right-hand  side  is  clearly  positive  and  so  dxfdO  is  positive  for  all 
real  values  of  6.  This  means  that  when  x  increases,  the  point  in  which  the 
stream  -line  meets  the  axis  of  x  moves  to  the  right  (i.e.  the  direction  in 
which  x  increases). 

If  we  increase  6  by  JTT,  F  (x)  is  transformed  into  —  G  (x),  and  if  we  add 
another  £77-  to  0,  the  function  —  G  (x)  is  transformed  into  —  F  (x),  conse- 
quently we  surmise  that  the  roots  of  F  (x)  =  0  are  separated  by  those  of 
G  (x)  =  0.  To  prove  this  we  adopt  Koshliakov's  method  of  proof  and 
calculate  the  derivative 

d  F  (x)  J  (x)  g'J^l^f  (x)  g  (x)  +  [F  (x)]*  +  [G  (*)]» 
dxG(x)~  ""  ~[~ 


This  is  clearly  positive  for  all  values  of  x  and  infinite,  perhaps,  at  the 


Koshliakov  *s  Theorem  225 

roots  of  O  (x)  =  0.    It  is  clear  from  a  graph  that  the  roots  of  F  (x)  are 
separated  by  those  of  0  (x)  =  0,  for  the  curve 


consists  of  a  number  of  branches  each  of  which  has  a  positive  shape. 

In  Koshliakov's  case  when  ks  —  0,  ms  =  1  the  functions  /  (x)  and  g  (x) 
are  polynomials  such  that  the  roots  of  the  equation  /  (x)  -f  ig  (x)  —  0  are 
of  type  zs  =  as  -f-  ibs,  where  bs  >  0.  The  associated  equation  F  (x)  =  0  is 
now  of  a  type  which  frequently  occurs  in  applied  mathematics.  In  par- 
ticular.if  /(;«;)  =  &&-*»,  p  (*)  =  (ft  +  ft)  a, 

the  roots  of  the  equation  /  (x)  -f  ig  (x)  =  0  are  ifa  and  i/32  and  so  we  have 
the  result  that  if  j8x  and  /?2  are  both  positive,  the  roots  of  the  equation 
(&L  +  &)  x  cos  (x  -  6)  -f  (&&>  -  #2)  sin  (a:  -  0)  =  0    ......  (B) 

are  all  real  and  increase  with  6. 

The  theorem  may  be  applied  to  the  cable  equation  by  writing  this  in 
the  form  >  ,  , 

7  _  2/^  fro  ±  Vl  ~   M2  (\  +   Al)} 

^  (r«  ~ 


where  y0(70  =  O,     y^C^  =  C,     L0  == 

Now 


t-  n  - 

=  (y0  +  2t>  - 

and  when  the  expression  on  the  right  is  equated  to  zero,  the  resulting 
algebraic  equation  for  p  has  roots  of  type  a  -t-  ib,  where  b  is  positive,  hence 
Koshliakov's  theorem  may  be  applied  and  the  conclusion  drawn  that  yil 
is  real.  Since  in  the  present  case  /x2  =  CLn2,  the  corresponding  value  of 
n  is  also  real. 

When  0  =  0,  x  =  wl,  /?,  =  j82  =  ZA,  the  equation  (B)  becomes  identical 

with  the  equation         »7              7       ,    0      -,„.    .       7  ,~x 

2/£o>  cos  o>&  —  (a)'2  — /i^)  «in  o>^,  (C) 

which  occurs  in  the  theory  of  the  conduction  of  heat  in  a  finite  rod,  when 
there  is  radiation  at  the  ends,  into  a  medium  at  zero  temperature. 
The  equations  of  this  problem  are  in  fact 

dv          S^v 
di  ~~      dx2' 
v=--f  (x),     for  t  =  0, 

—  ~  +  hv  =  0  at  x  =  0,     =-  +  hv  =  0  at  x  =  I, 
ox  ox 

and  are  satisfied  by 

v  =  e-K<°2t  [A  cos  a>x  -f  B  sin  o>#] 

if  -  wB  +  hA  -  0, 

a>  (B  cos  a)l  —  A  sin  o>Z)  +  h(B  sin  toZ  -f  -4  cos  o>Z)  =  0. 

B  15 


226  Two-dimensional  Problems 

Eliminating  A/B  the  equation  (C)  is  obtained.  The  problem  is  finally 
solved  by  a  summation  over  the  roots  of  this  equation,  the  root  w  =  0 
being  excluded. 

Equations  similar  to  (A)  occur  in  other  branches  of  physics  and  many 
useful  analogies  may  be  drawn.  In  the  theory  of  the  transverse  vibrations 
of  a  string  we  may  suppose  that  the  motion  of  each  element  of  the  string 
is  resisted  by  a  force  proportional  to  its  velocity*.  The  partial  differential 
equation  then  becomes 


- 
~ 


which  is  of  the  same  form  as  (A)  if  K  =  RjL,  c2  =  l/LC. 

An  equation  of  the  same  type  occurs  also  in  Rayleigh's  theory  of  the 
propagation  of  sound  in  a  narrow  tube,  taking  into  consideration  the 
influence  of  the  viscosity  of  the  mediumf. 

Let  X  denote  the  total  transfer  of  fluid  across  the  section  of  the  tube 
at  the  point  x.  The  force,  due  to  hydrostatic  pressure,  acting  on  the  slice 
between  x  and  x  +  dx,  is 

0  dp  .       2  ,  d*x 

—  8  -if-  ax  =  a2pdx  -*—<,  , 
dx  r       dx2 

where  8  is  the  area  of  the  cross-section,  p  is  the  pressure  in  the  fluid,  p  is 
the  density  and  a  is  the  velocity  of  propagation  of  sound  waves  in  an 
unlimited  medium  of  the  same  material. 

The  force  due  to  viscosity  may  be  inferred  from  the  investigation  for 
a  vibrating  plane  (§  3-15),  provided  that  the  thickness  of  the  layer  of  air 
adhering  to  the  walls  of  the  tube  be  small  in  comparison  with  the  diameter. 
Thus,  if  P  be  the  perimeter  of  the  inner  section  of  the  tube  and  F  the 
velocity  of  the  current  at  a  distance  from  the  walls  of  the  tube,  the  tan- 
gential force  on  a  slice  of  volume  Sdx  is,  by  the  result  of  (§3-15,  Ex.  1), 
equal  to 


where  n/27r  is  the  frequency  of  vibration. 

o  v- 

Replacing  VS  by  -~r  we  can  say  that  the  equation  of  motion  of  the 

ut 

fluid  for  disturbances  of  this  particular  frequency  is 


- 

or  S 


, 

=  a2  -«--,  . 
cte2 

*  Rayleigh,  Theory  of  Sound,  vol.  I,  p.  232.  t  Ibid.  voL  n,  p.  318, 


Air  Waves  in  Pipes  227 

This  equation  has  been  used  as  a  basis  for  some  interesting  analogies 
between  acoustic  and  electrical  problems*.  We  shall  write  it  in  the  abbre- 
viated form 


—- 

dt*  dt  ~       dx*' 

Rayleigh's  equation  has  been  used  recently  by  L.  F.  G.  Simmons  and 
F.  C.  Johansen  in  a  discussion  of  their  experiments  on  the  transmission  of 
air  waves  through  pipesf. 

At  the  end  x  =  0  the  boundary  condition  is  taken  to  be 

X  =  X0sm(nt),  ......  (D) 

and  a  solution  is  built  up  from  elementary  solutions  of  type 

X  =  Aetnt±mx, 
where  a2ra2  =  —  Hn2  -f  iKn. 

Since  m  is  complex,  we  write  m  =  a  4-  ip.   A  solution  appropriate  for 

o  y- 

a  pipe  of  length  I  with  a  free  end  (x  =  I)  at  which  x  -  =  0  is 

X  =  A  {e~ax  sin  (nt  -  px)  +  e-**'  sin  (nt  -  fix')} 

+  C  {e-**  cos  (nt  -  px)  -f  e~ax'  cos  (nt  -  px')}, 

where  xf  =  21  —  x,  and  where  the  constants  A,  C  are  chosen  so  that 
X0  =  A  {1  +  e-2*'  cos  2  pi}  -f  Ce-™  sin  2ply 

0  =  -  Ae-**1  sin  2pl  +  G  {1  4-  e~*al  cos  2  pi}  . 
These  equations  give 

^r  =  (1  +  e~™  cos  2pl)  XQ,       G  -  e-*1  sin  2pl  .  Z0, 
where  T  =  1  -f  2e~2aZ  cos  2j8/  -f  e~^1. 

In  the  case  of  a  pipe  with  a  fixed  end  the  boundary  condition  is  X  =  0 
at  #  =  Z,  anc  we  write 

X  =  A  [>-«*  sin  (w<  -  px)  -  e-**'  sin  (^  -  px')] 

-f  (7  [e~aa!  cos  (nt  -  £#)  -  e—*'  cos  (n^  -  j8a?')]- 
The  boundary  condition  (D)  is  now  satisfied  if 

Z0  =  A  {1  -  e-2*'  cos  2pl}  -  Ce~™  cos  2j8«, 
0  =  Ae~™  sin  2j8Z  +  G  {1  -  e-2*'  cos 


Therefore 

G^^L  =  (1  -  e-**  cos  2^)  Jf09      GC7  =  -  XQe-**  sin  2)81, 
where  G  =  1  -  2e"2ttZ  cos  2^8Z  -f-  e^1. 

*  See  a  recent  discussion  by  W.  P.  Mason  in  the  Bell  System  Technical  Journal,  vol.  vi,  p.  258 
(1927). 

t  Advisory  Committee  far  Aeronautics,  vol.  n,  p.  661  (1924-5)  (E.-M.  957,  Ae.  176). 

15-2 


228  Two-dimensional  Problems 

If  y  denotes  the  ratio  of  the  specific  heats  for  air,  the  pressure  at  any 
point  exceeds  the  normal  pressure  p0  by  the  quantity 


where  X  =  £S.  The  excess  pressure  at  the  fixed  end  is  consequently 

P-Po  =  2&«~"  YPo  («2  +  £»)*  °~l  sin  (nt  ~  &  +  #). 
.  .        ,       pA  +  aC 

where  tan  ^  =  a^  T^' 

The  following  conclusion  is  derived  from  a  comparison  of  theory  with 
experiment  : 

"Marked  divergence  between  observed  and  calculated  results  shows 
that  existing  formulae  relating  to  the  transmission  of  sound  waves  through 
pipes  cannot  be  successfully  employed  for  correcting  air  pulsations  of  low 
frequency  and  finite  amplitude." 

§  3-21.  Vibration  of  a  light  string  loaded  at  equal  intervals.  In  recent 
years  much  work  has  been  done  on  methods  of  approximation  to  solutions 
of  partial  differential  equations  by  means  of  a  method  in  which  the  partial 
differential  equation  is  replaced  initially  by  a  partial  difference  equation 
or  an  equation  in  which  both  differences  and  differential  coefficients  appear. 
Such  a  method  is  really  very  old  and  its  first  use  may  be  in  the  well-known 
problem  of  the  light  string  loaded  at  equal  intervals.  This  problem  was 
discussed  by  Bernoulli*  and  later  in  greater  detail  by  Lagrange|. 

Let  the  string  be  initially  along  the  axis  of  x  and  let  the  loading  masses, 
which  we  assume  to  be  all  equal,  be  concentrated  at  the  points 

x  =  na,     n  =  0,  ±  1,  ±  2,  ____ 

Let  yn  be  the  transverse  displacement  in  a  direction  parallel  to  the 
«/-axis  of  the  mass  originally  at  the  point  na,  then  if  the  tension  P  is  re- 
garded as  constant,  we  have  for  the  motion  of  the  nth  particle 

amyn  =  P  (yn+l  -  yn)  +  P  (yn_^  -  yn). 
Writing  k2am  =  P,  the  equation  becomes 

tin  -  &  (yn+1  +  yn_i  -  2yn).  ......  (A) 

Let  us  now  put      uzn  =  yn,  u2n+1  =  k(yn-  yn+1), 
then  ii2n  =  k  (u2n^  -  u2n+1), 

or,  if  s  is  any  integer, 

*  Johann  Bernoulli,  Petrop.  Comm.  t.  m,  p.  13  (1728);  Collected  Works,  vol.  in,  p.  198. 
t  J.  L.  Lagrange,  Me'canique  Analytique,  1.  1,  p.  390. 


Vibration  of  a  Loaded  String  229 

This  is  a  difference  equation  satisfied  by  the  Bessel  functions  and  a 
particular  solution  which  will  be  found  useful  is  given  by* 

us  =  AJ8^  (2kt), 
where  A  and  a  are  arbitrary  constants  and 

oo     (__\s  (l~\m+2s 

'•M-.?.*-.-!  nff.+  D <B) 

Let  us  first  consider  the  ideal  case  of  an  endless  string  and  suppose  that 
initially  all  the  masses  except  one  are  in  their  proper  positions  on  the  axis 
of  x  and  have  no  velocity,  while  the  particle  which  should  be  at  x  =  na 
has  a  displacement  yn  =  ^  and  a  velocity  yn  =  v,  then  the  initial  conditions 

are  7  7 

u2n  =  v,      u2n+l  -  £77,       u2n_i  =•  -  £??, 

while  us  is  initially  zero  if  s  does  not  have  one  of  the  three  values  2n  —  1, 
2/r,  2n  -f  1.   A  solution  which  satisfies  these  conditions  is 

us  =  v  JS_2M  (2fc)  +  fey  [Js-2n-i  (2fo)  -  e/5_2n+1  (2fa)L 

for,  when  t  =  0,  JT  (2fe)  is  zero  except  when  r  =  0  and  then  the  value  is 
unity. 

When  all  the  masses  have  initial  velocities  and  displacements  the 
solution  obtained  by  superposition  is 

us  =  XvnJs_2n  (2kt)  -f  k^n  [J^^  (2kt)  -  J8_2n+l  (2kt)] (C) 

If  vn  =  0  we  find  by  integration  that 

ys  =  Si,n  J2s_2n  (2fcJ).  (D) 

Let  us  now  discuss  the  case  when  this  series  reduces  to  one  term, 
namely,  the  one  corresponding  to  n  =  0.  Referring  to  the  known  graph 
of  the  function  J2s  (2kt),  to  known  theorems  relating  to  the  real  zeros  and 
to  the  asymptotic  representation  f 

J28  (2kt)  =  (7r&)-*cos(2  to  -  -^-j-  »).  (E) 

we  obtain  the  following  picture  of  the  motion : 

The  disturbed  mass  swings  back  into  its  stationary  position,  passes 
this  and  returns  after  reaching  an  extreme  position  for  which  |  y  |  <  ^ . 
Its  motion  always  approaches  more  and  more  to  an  ordinary  simple 
harmonic  motion  with  frequency  initially  greater  than  &/TT,  but  which  is 
very  close  to  this  value  after  a  few  oscillations.  The  amplitude  gradually 
decreases,  the  law  of  decrease  being  eventually  (rrkt)"^.  This  diminution 

*  T.  H.  Havelock,  Phil.  Mag.  (6),  vol.  xix,  p.  191  (1910);  E.  Schrodingor,  Ann.  d.  Phys. 
Ed.  XLIV,  S.  916  (19H);  M.  Koppo,  Pr.  (No.  96)  Andreas- Realgymn.  Berlin  (1899),  reviewed 
in  Fortschritte  der  Math.  (1899). 

f  Whittaker  and  Watson,  Modern  Analysis,  p.  368.  The  formula  is  due  to  Poisson.  An  ex- 
tension of  the  formula  is  obtained  and  used  by  Koppe  in  his  investigation.  The  complete  asymptotic 
expansion  is  given  in  Modern  A  nalysis. 


230  ,         Two-dimensional  Problems 

depends  on  the  fact  that  the  vibrational  energy  of  the  mass  is  gradually 
transferred  to  its  neighbours,  which  part  with  it  gradually  themselves  and 
so  on  along  the  string  in  both  directions.  After  a  long  time,  when  2kt  is 
so  large  that  the  asymptotic  representation  (E)  can  be  used  for  the 
Bessel  functions  of  low  order,  the  masses  in  the  neighbourhood  of  the 
origin  vibrate  approximately  in  the  manner  specified  by  the  "limiting 
vibration"  of  our  arrangement,  neighbouring  points  being  in  opposite 
phase.  The  amplitudes,  however,  decrease  according  to  the  law  mentioned 
above  and  the  range  over  which  this  approximate  description  of  the 
vibration  is  valid  gets  larger  and  larger. 

According  to  the  formula  (D)  all  the  masses  are  set  in  motion  at  the 
outset,  and  all,  except  the  one  originally  displaced,  begin  to  move  in  the 
positive  direction  if  T?O  >  0. 

Let  us  consider  the  way  in  which  the  mass  originally  at  x  =  na  begins 
its  motion.  The  larger  n  is,  the  slower  is  the  beginning  of  the  motion  and 
the  longer  does  it  continue  in  one  direction.  This  is  because  J2n  (2kt) 
vanishes  like  Ant2H,  as  /  approaches  zero,  An  being  the  constant  multiplier 
in  the  expansion  (B).  Also  because  the  first  value  oi  2kt  for  which  the  func- 
tion vanishes  lies  between  ^/{(2ri)  (2n  -f  2)}  and  V{(2)  (%n  +  *)  (2^  +  3)}. 

It  is  interesting  to  note  that  in  this  elementary  disturbance  there  is 
no  question  of  a  propagation  with  a  definite  velocity  c  as  we  might  expect 
from  the  analogous  case  of  the  stretched  string.  Let  us,  however,  examine 
the  case  in  which  all  the  particles  are  set  in  motion  initially  and  in  such  a 
way  that  the  resulting  motion  is  periodic. 

Writing  yn  =  Yne2lku>t  we  have  the  difference  equation 


If  a)  =  sin  <f>  this  equation  is  satisfied  by 

Yn  =  A  sin  2n<f>  +  B  cos  2n<f>.  ......  (F) 

Choosing  the  particular  solution 

Y   =  Ce~2in* 

•*•  n  —  ^c  9 

we  have  yn  =  Ce2i(ktB{n*~n*}. 

Making  kt  sin  <£  —  n<j>  constant  we  see  that  the  phase  velocity  is 

_  ak  sin  <f> 

«-—?—• 

The  period  T  is  given  by  the  equation 

T  =  —  __ 
k  sin  </>  ' 

and  the  wave-length  A  by  the  equation 

A  =  cT  «  ira/<f>. 


Group  Velocity  231 

The  phase  velocity  thus  depends  on  the  wave-length  and  so  there  is  a 
phenomenon  analogous  to  dispersion.  Introducing  the  idea  of  a  group 

velocity  U  such  that  ~x          ^, 

3A      r/9A 

3*  +  Udx~^ 

that  is,  such  that  A  does  not  vary  in  the  neighbourhood  of  a  geometrical 
point  travelling  with  velocity  U,  we  next  consider  a  geometrical  point 
which  travels  with  the  waves.  For  this  point  A  varies  in  a  manner  given 

by  the  equation  *  ^         ~^          ~  ,    a 

^        ^A  _  x  dc  _  ,  dc  cc 

a7  +  Cfo         fo         5Aai' 

the  second  member  expressing  the  rate  at  which  two  consecutive  wave- 
crests  are  separating  from  one  another.  Eliminating  the  derivatives  of  A 
we  obtain  the  formula  of  Stokes  and  Rayleigh, 

Z7  =  c-A~=tf(A),say: 

In  the  present  case 

U  =  ak  cos  (7ra/A)  =  ak  cos  <f>. 

When  A->  oo,  U ->  ak  =  U(co)  =  c(oo). 

Hence  for  long  waves  the  group  velocity  is  approximately  the  same  as 
the  wave  velocity.  For  the  shortest  waves  <f>  =  \TT,  we  have  U  =  0. 

When  there  are  only  n  masses  the  two  extreme  ones  being  at  distance 
a  from  a  fixed  end  of  the  string,  the  equations  of  motion  are 

&  +  &  (2ft  -  0  -  ft)  =  0, 
$2  +  &  (2y2  -  ft  -  ft)  =  0, 


Assuming  ys  =  Y8e2ik<at  as  before  and  eliminating  the  quantities  Y8 
from  the  resulting  equations  we  obtain  the  following  condition  for  free 
oscillations : 

2  cos  2<f>        -  1  0  0     |  =  0, 

-  1        2  cos  2^        —  1          0 
0  -  1        2  cos  2<j>     -  1 

t 

where  there  are  n  rows  and  columns  in  the  determinant.  Since 


it  is  readily  shown  by  induction  that 


n  ~         sin  2<f> 
*  See  Lamb's  Hydrodynamics,  p.  359. 


232  Two-dimensional  Problems 

This  is  zero  if  2  (n  -f-  1)  (/>  =  rvr,  r  =  1,  2,  ...  n;  we  thus  obtain  /i  different 
natural  frequencies  of  vibration.  When  the  motion  corresponding  to  any 
one  of  these  natural  frequencies  is  desired  we  use  an  expression  of  type 
(F)  for  Yn  and  the  end  condition  Yn  =  0  will  be  satisfied  by  writing  B  =  0. 
Hence  one  of  the  natural  vibrations  is  given  by 

ym  =  A  sin2w<£e2i*<8ln*, 

where  2  (n  +  1)  (f>  =  TTT     (r  =  1,  2,  ...  n). 

If  the  velocity  is  initially  zero  we  write 

ys  =  ^4  sin  2s<^  .  cos  (2&£  sin  <f>). 

Let  us  examine  more  fully  the  case  in  which  n  =  2.  The  possible  values 
of  (f>  are  and  -- ,  consequently  in  the  first  case 


A  sin  ^  cos  (2Kt  sin  ^  j ,       y2  =  A  sin  ~  cos  ( 2kt  sin  ~  j ; 


yl  and  j/2  have  the  same  sign  and  the  string  does  not  cross  the  axis  of  x. 
In  the  second  case 

A       .      2?!  /rt/J      .       77\  ,.477  /rt/J      .       77 

yl  =  A  sm       cos  (  2tct  sin  -   ,      s/2  =  A  sin  —  cos  (  2kt  sm  - 

o  \  o/  o  \  o 

^  and  y2  have  opposite  signs,  the  string  crosses  the  axis  of  z  at  its  middle 
point  which  is  a  node  of  the  vibration. 

When  n  =  3  we  find  in  a  similar  way  that  there  is  one  vibration  without 
a  node,  one  with  a  node  and  one  with  two  nodes. 

The  extension  to  the  case  in  which  n  has  any  integral  value  is  clear. 
The  general  vibration,  moreover,  is  built  up  by  superposition  from  the 
elementary  vibrations  which  have  respectively  0,  1,  2,  ...  n  —  1  nodes,  the 
nodes  of  one  elementary  vibration  such  as  the  5th  being  separated  by  those 
of  the  (s  -  l)th. 

If  we  regard  this  solution  as  valid  for  all  integral  values  of  s,  we  may 
apply  it  to  the  infinite  string.  The  initial  value  of  ya  is  now  A  sin  2s<f>  and 
so  by  applying  the  general  formula  we  are  led  to  the  surmise  that  there  is 
a  relation 

00 

sin  2s<f> .  cos  (2kt  sin  cf>)  ==    S    sin  2p<f> .  J2s-2»  (2&0> 

J>=  —00 

which  is  true  for  all  real  values  of  <f>.  This  relation  is  easily  proved  with 
the  aid  of  well-known  formulae. 

An  equation  similar  to  (A)  occurs  in  the  theory  of  the  vibrations  of  a 
row  of  similar  simple  pendulums  (a,  m)  whose  bobs  are  in  a  horizontal 
line  and  equally  spaced,  consecutive  bobs  being  connected  by  springs  as 
shown  in  Fig.  14.  Using  yn  to  denote  the  horizontal  deflection  of  the  nth 


Electrical  Filter  233 

bob  along  the  line  of  bobs  and  supposing  that  the  constants  of  the  springs 
are  all  equal,  the  equations  of  motion  are  of  type 

/  /  x       7  /  x      my  //-.x 

tYlij     ==  K  (I/         I/   }  rC  ( I/     11        ) —  I/  (\JTI 

The  periodic  solutions  of  this  equa-  

tion  give  a  good  illustration  of  the  filter 
properties  of  chains  of  electric  circuits 
that  were  discovered  by  G.  A.  Camp- 
bell*. The  mechanical  system  may,  in 
fact,  be  regarded  as  an  analogue  of  the  Flg>  14' 

following  electrical  system  consisting  of  a  chain  of  electrical  circuits  each 
of  which  contains  elements  with  . 

inductance  and  capacitance  (Fig.       I      '  .  . 

15).  _-   v/n-i  — r—    (*n    — T—   V^TIM 

The    following    discussion    is  — ' ' ' 

based  largely  upon  that  of  T.  B.  Fig' 15' 

Brown  f.   When  the  chain  is  of  infinite  length  and  the  motion  is  periodic 

$n  appropriate  solution  is  obtained  by  writing 

yn  =  Ar~n  sin  (pt  —  n<f>). 

If  mg  —  ap2  =  2akQ  the  equations  for  r  and  (f>  are 

(1  -  r2)  sin  <f>  =  0,       (r2  +  1)  cos  <f>  =  2r  (1  +  Q). 

These  are  satisfied  by  <f>  =  0,  r  ^  1  and  by  r  =  1,  cos  <f>  =  1  +  Q.  In  the 
latter  case  there  is  transmission  without  attenuation  but  with  a  change  of 
phase  from  section  to  section,  the  phase  velocity  corresponding  to  a  fre- 
quency /  =  p/2n  being  v^px  =  Zirfx 

(h         (b 

where  x  is  the  length  of  each  section.  This  type  of  transmission  is  possible 
only  when  Q  lies  between  0  and  -  2,  that  is  when/ lies  between /t  and/2, 
where  ,  .  I/AJ  \ 

/  /  fl\  I    4/c       Q  \ 

O— f  I  i  &  1  9^-f    /  I  _L  y    I 

ZTT/I  =  A  /  I  -  I ,      ^^[/2  ~~  A  /  I         '    "  I  • 
J1      V  \aJ  V  \w     a/ 

This  range  of  frequencies  gives  a  pass  band  or  transmission  band.  On 
the  other  hand,  when  <£  =  0  we  have  r  =  1  -f  Q  ±  [(1  +  Q)2  -  1]*,  and  it 
is  clear  that  r  >  0  when  /<  fl9  r  <  0  when  />  /2.  The  negative  value  of 
r  indicates  that  adjacent  sections  are  moving  in  opposite  directions  with 
amplitudes  decreasing  from  section  to  section  as  we  proceed  in  one  direction 
down  the  line.  We  may  use  a  positive  value  of  r  if  we  take  <f>  =  TT  instead 
of  <f>  =  0. 

It  should  be  noticed  that  r  is  real  only  when  /  lies  outside  the  pass 

*  U.S.  Patent  No.  1,227,113  (1917);  Bell  System  Tech.  Journ.  p.  1  (Nov.  1922). 
t  Spurn.  Opt.  Soc.  America,  vol.  viu,  p.  343  (1924). 


234  Two-dimensional  Problems 

band.  There  are  two  regions  in  which  /  may  lie  and  these  are  called  stop 
bands  or  suppression  bands  ;  one  of  these  is  direct  and  the  other  reverse. 
The  stopping  efficiency  of  each  section  is  represented  by  loge  |  r  \  and  this 
is  plotted  against  /in  Brown's  diagram. 

For  a  further  discussion  of  wave-filters  reference  may  be  made  to 
papers  by  Zobel,  Wheeler  and  Murnaghan. 

Let  us  now  write  equation  (G)  in  the  form 

(D2  +  c2)  yn  -  &2  (yM  +  yn-1)f 

where  b*=k,      c2  =  ~+?,     D=~4, 

m  ma  at 

and  let  us  seek  a  solution  which  satisfies  the  initial  conditions 

2/0=1,       * 
2/o  =  0*       ?/i 

One  way  of  finding  the  desired  solution  is  to  expand  yn  in  ascending 
powers  of  62.  Writing 

yn  =  (n,  n)  b*n  +  (n,n  +  2) 
it  is  found  by  substitution  in  the  equation  that 

262    \n     (/&_+_!)  (n  +  2)  /_J  262 
~       ~ 


1   ["/     262    \n 
2/71  ~~  2»  [U>2  +  c2  / 


_ 

2  .  4(2n  +  2)T2rT+  4) 

the  law  of  the  coefficients  being  easily  verified.  The  meaning  which  must 
be  given  to  2 


is  one  in  which  the  Taylor  expansion  of  the  expression  in  powers  of  t  starts 
with  t2m  .  (2b2)m/(2m)  I 

An  expression  which  seems  to  be  suitable  for  our  purpose  is  obtained 
by  writing 

cos  ct  =  75—      ezt  ~*- 


where  C  is  a  circle  with  its  centre  at  the  origin  and  with  a  radius  greater 
than  c.  The  result  of  the  operation  is  then 

2b2      m  I  zdz 


and  we  obtain  the  formal  expansion 

-  I  _1_  f  *t  _  ldz_  [7  262  V  4.  (»+1)(ra+2)  /_  2^a_V^  ,     I 

y"      2"  '  2wi  Jce    z2  +  c2  LV32  +  cV          2  (2»  +  2)     Ua  +  cV  ;'  +  '"J  ' 


Torsional  Vibrations  of  a  Shaft  235 

In  particular, 

1     r  eztzdz 

~ 


eztzdz 


1     r 

2/1  =  2*ri  ]e  [>2c~  464}*  z2  +  c2  -f  [(z2+c2)2- 
and  generally 


1     r 
"  ^  2m  j 


2fe2 


!  c  [(z2  +  c2)2  -  464]i  (z2  +  c2  +  [(z2  +  c2)2  -  46*]*)    * 

It  is  easy  to  verify  that  this  expression  satisfies  equation  (G)  and  the 
prescribed  initial  conditions. 

When  c2  =  26 2  the  formula  reduces  to 

!_  f          e"  _     \ b  ___ 


which  must  be  equivalent  to  J2n  (2bt). 
The  solution  of  the  equation 

(D  +  c2)  yw  =  62  (^^  +  y,^), 
which  satisfies  the  initial  conditions 

2/o  =  1,     2/i  =  0,     t/2  =  0,  ...         when  t  =  0, 
is  given  by  the  formula 

=  ^  f          _e?*zdz 


An  equation  which  is  slightly  more  general  than  (A)  occurs  in  the  theory 
of  the  torsional  vibrations  of  a  shaft  with  several  rotating  masses*.  This 
theory  can  be  regarded  as  an  extension  of  that  of  §  1-54  and  as  a  preliminary 
study  leading  up  to  the  more  general  case  of  a  shaft  whose  sectional  pro- 
perties vary  longitudinally  in  an  arbitrary  manner. 

Let  /!,  /2,  /3,  ...  /jv  be  the  moments  of  inertia  of  the  rotating  masses 
about  the  axis  of  the  shaft,  019  02,  03,  .  .  .  0N  the  angles  of  rotation  of  these 
masses  during  vibration,  kl9  k2,  k3,  ...  kN  the  spring  constants  of  the  shaft 
for  the  successive  intervals  between  the  rotating  masses.  Then 

*i  (*i  ~  02),  ^2  (<?2  -  9*),  -.  *W  (Vi  -  ON) 

are  torque  moments  for  these  intervals.  Neglecting  the  moments  of  inertia 
of  these  intervening  portions  of  the  shaft  in  comparison  with  /x  ,  72  ,  ...  /# 
the  kinetic  energy  T  and  the  potential  energy  V  of  the  vibrating  system  are 
given  by  the  equations 

2T=/1d1*+/A"+.../A>, 

2F  =  k,  (0,  -  02)*  +  kt  (02  -  03)2  +  -  **-i  (0.v-i  -  »*)', 

*  See  S.  Timoshenko,  Vibration  Problems  in  Engineering,  p.  138;  J.  Morris,  The  Strength  of 
Shafts  in  Vibration,  ch.  x  (Crosby  Lockwood,  London,  1929). 


236  Two-dimensional  Problems 

and  Lagrange's  equations  give 

/A  +  kn  (0n  -  0n+1)  -  kn_,  (0^  -  0n)  =  0, 

Except  for  the  two  end  equations,  for  which  n  =  I,  N,  respectively, 

these  equations  are  the  same  as  those  of  a  light  string  loaded  at  unequal 

'Intervals.  When  k^  -----  Jc2  =  &3  =  ...  =  kN  the  foregoing  analysis  can  be  used 

with  slight  modifications.  A  second  case  of  some  mathematical  interest 

arises  when  A^  =  Jfc3  =  fe6  =  ...  fc 

7.     __    7*    __    7^     __  7, 

^ 2  —       4  ~~       6    ~~    •••   '^* 

EXAMPLES. 

1.    By  considering  special  solutions  of  the  equation  of  the  loaded  string  prove  that  the 
following  relations  are  indicated: 

00 

(n  -  x?  =      S 

7/1=  -CO 


2.  Prove  that  the  equation 

J  M 

^  =  J  [^_,  (x)  +  ^n+1  (x)  -  2^n  (a:)] 
is  satisfied  by  Fn  (x)  =  e^t*^)       2      7n_m  (*  -  a)  Fm  (a), 

m^-~<x> 

where  /n  (a-)  =  *~Vn  (ix), 

and  obtain  the  solution  in  the  form  of  a  contour  integral. 

3.  Prove  that  the  equation 

lln  =  V  [yn+z  -H  2/n_2 
is  satisfied  by 

1     (          _  e 
?w/n  "  2^J  Jc  [(Za~+  c2)2  -  46*1     z2  +  c-f  [(z2  +  c2)2- 

4.  Each  mass  in  a  system  is  connected  with  its  immediate  neighbours  on  the  two  sides 
by  elastic  rods  capable  of  bending  but  without  inertia.  Assuming  that  the  potential  energy 
of  bending  is 

F  =  . 


prove  that  the  oscillations  of  the  system  are  given  by  an  equation  of  type 


when  r  >  1  and  obtain  the  two  end  equations.  [Lord  Rayleigh,  Phil.  Mag.  (5),  vol.  XLTV,  p.  356 
(1897);  Scientific  Papers,  vol.  rv,  p.  342.] 

6.   Prove  that  a  solution  of  the  last  equation  is  given  by 

Vr  =  Jr  Q  C08  (Y  ""  !•)  *  [Havelock.] 

§3-31.   Potential  function  with  assigned  values  on  a  circle.    Let  the 
origin  and  scale  of  measurement  be  chosen  so  that  the  circle  is  the  unit 


Potential  with  Assigned  Boundary  Values  237 

circle  |  z  \  =  1  and  let  z'  =  eie/  be  the  complex  number  for  a  point  P'  on 
this  circle.  Our  problem  is  to  find  a  potential  function  V  which  satisfies 
the  condition  T7 


as  z~*  z  - 

To  make  the  problem  more  precise  the  way  in  which  the  point  z 
approaches  z'  ought  to  be  specified  and  something  must  be  said  about  the 
restrictions,  if  any,  which  must  be  laid  on  the  function/  (0').  These  points 
will  be  considered  later;  for  the  present  it  will  be  supposed  simply  that 
/  (0')  is  real  and  uniquely  defined  for  each  value  of  0'  when  9'  is  a  real  angle 
between  —  TT  and  rr.  The  mode  of  approach  which  will  be  considered  now 
is  one  in  which  z  moves  towards  z'  along  a  radius  of  the  unit  circle.  In 
other  words,  if  z  =  re!*,  where  r  and  9  are  real,  we  shall  suppose  that  9 
remains  equal  to  9'  and  that  r  ->  1. 

Now  let  z  =  r  .  e~id  be  the  complex  quantity  conjugate  to  z;  an  attempt 
will  be  made  first  of  all  to  represent  V  by  means  of  a  finite  or  infinite  series 

F0  =  c0  +  2  (cnzn  +  c^i"),  ......  (A) 

71-1 

where  c0  is  a  real  constant  and  cn9  c_n  are  conjugate  complex  constants. 
When  the  series  contains  only  a  finite  number  of  terms  it  evidently 
represents  a  potential  function  and  in  the  limiting  process  z-+z'  it  tends 
to  the  value  Ecnz'n,  where  the  summation  extends  over  all  integral  values 
of  n  for  which  cn  ^  0.  Negative  indices  are  included  because  zn  ->  z'~n. 

Supposing  now  that  the  finite  series  represents  the  function/  (#'),  the 
coefficient  cn  is  evidently  given  by  the  formula 


for  the  integral  of  eim6'  between  —  -n  and  TT  is  zero  unless  m  =  0,  conse- 
quently the  term  cnz'n  in  the  series  for/  (#')  is  the  only  one  which  contributes 
to  the  value  of  the  integral. 

A  function/  (0')  which  can  be  represented  as  the  sum  of  a  finite  number 
of  terms  of  type  cne~ine'  is  evidently  of  a  special  nature  and  the  natural 
thing  to  do  is  to  endeavour  to  extend  the  solution  which  has  just  been  found 
by  considering  the  case  in  which  an  infinite  number  of  the  constants  cn 
defined  by  the  formula  (B)  are  different  from  zero.  The  series  (A)  formed 
from  these  constants  then  contains  an  infinite  number  of  terms. 

Let  us  now  assume  that  the  f  unction  /(#')  is  integrable  in  the  interval 
—  TT  <  6'  <  IT.  Since  the  series 


1  +  S  rn  [e*»<*-*'>  +  e-in<'-6'>]  ==  K  (9  -  9') 
i 

is  uniformly  convergent  for  all  points  of  this  interval  if  |  r  |  <  1,  it  may  be 


238  Two-dimensional  Problems 

integrated  term  by  term  after  it  has  been  multiplied  by/  (0').  The  potential 
function  V  may,  consequently,  be  expressed  in  the  form 

d0'9  (C) 


where          K  (co)  =1  +  2  £  rn  cos  na>  =  -  —  ~  —  ---  A. 

n-i  1  -  2r  coso>  -f  r2 

The  integral  representing  our  potential  function  F  is  generally  called 
Poisson's  integral  and  will  be  denoted  here  by  the  symbol  P  (r,  6)  to 
indicate  that  it  depends  on  both  r  and  6.  This  integral  is  of  great  importance 
in  the  theory  of  Fourier  series  as  well  as  in  the  theory  of  potential  functions. 

The  formula  (C)  may  be  obtained  in  another  way  by  using  the  Green's 
function  for  the  circle.  If  P  (r,  6)  is  the  pole  of  the  Green's  function, 
Q  (r-1,  6)  the  inverse  point  and  P'  (/,  6')  an  arbitrary  point  which  is  inside 
the  circle  (or  on  the  circle)  when  P  is  inside  the  circle  and  outside  (or  on) 
the  circle  when  P  is  outside  the  circle,  an  appropriate  expression  for  the 
Green's  function  is 


where  A  is  an  arbitrary  point  on  the  circle.  This  expression  is  evidently 
zero  when  P'  is  on  the  circle,  it  becomes  infinite  in  the  desired  manner  when 
P'  approaches  P  and  it  is  evidently  a  potential  function  which  is  regular 
except  at  P. 
The  formula 

"  2~7r  J  o    F-  2r  cos~(0  ~(F)+~r* 

represents  a  potential  function  which  takes  the  value  /  (0)  on  the  circle 
and  is  regular  both  inside  and  outside  the  circle. 
When  r  =  0  the  formula  gives  the  relation 


where  F0  is  the  value  of  F  at  the  centre  of  the  circle.  This  is  the  two- 
dimensional  form  of  Gauss's  mean  value  theorem. 

When  /  (6)  is  real  for  real  values  of  6  the  formula  (0)  may  be  written 
in  the  form 


of  the  sum  of  two  conjugate  complex  quantities  each  of  which  takes  the 
value  J/  (6)  at  the  point  z  =  eie  on  the  unit  circle,  and  we  deduce  Schwarz's 
more  general  expression 


......  (D) 


Potential  with  Assigned  Boundary  Values  239 

for  a  function  F  (z)  whose  real  part  on  the  unit  circle  is/  (6).  The  imaginary 
part  of  F  (z)  is  a  potential  function  i  W  which  is  given  by  the  formula 

w  -  h  4-  1  (**rsm(e-6')f(e')  M' 
0      TrJo    l-2reos(0-0')  +  r2' 

If  in  the  formula  (D)  we  have  /  (2-n  —  6)  =  /  (0)  we  obtain  Boggio's 
formula  for  a  function  F  (z)  whose  real  part  takes  an  assigned  value  /  (0) 
on  the  semicircle  z  =  eie,  0  <  9  <  TT,  and  whose  imaginary  part  vanishes 
on  the  line  z  =  cos  a,  0  <  a  <  TT,  i.e.  the  diameter  of  the  semicircle, 


1-  2zcos0'-f  z2* 

When  F  =  /  (z),  where  /  (z)  is  a  function  which  is  analytic  in  the  unit 
circle  |  z'  —  z  \  —  1,  Gauss's  mean  value  theorem  may  be  written  in  the 
form 


and  is  then  a  particular  case  of  Cauchy's  integral  theorem.  By  means  of 
the  substitution  z'  =  pz,  F(pz)--=f(z)  the  theorem  may  be  extended  to  a 
circle  of  radius  p. 

If  on  the  circle  we  have  |  /  (zf),  \  <  M,  the  formula  shows  that  |  /  (z)  |  <  M. 

More  generally  we  can  say  that  if  /  (z)  is  a  function  which  is  regular 
and  analytic  in  a  closed  region  G  and  is  free  from  zeros  in  G,  then  the 
greatest  value  of  |  /  (z)  \  is  attained  at  some  point  of  the  boundary  of  G 
and  the  least  value  of  |  /  (z)  \  is  also  attained  at  some  point  on  the  boundary 
of  G.  In  this  statement  values  of  /  (z)  for  points  outside  G  are  not  taken 
into  consideration  at  all. 

If/  (z)  is  constant  the  theorem  is  trivial.  If/  (z)  is  not  constant  and  has 
its  greatest  value  M  at  some  point  z0  inside  G  we  can  find  a  small  neigh- 
bourhood of  z0  entirely  within  G  for  which  |  /  (z)  \  <  M  ,  and  if  O  is  a  small 
circle  in  this  neighbourhood  and  with  z0  as  centre  this  inequality  holds  for 
each  point  of  C  and  so 


which  leads  to  a  contradiction.  The  theorem  relating  to  the  minimum  value 
of  |  /(z)  |  may  be  derived  from  the  foregoing  by  considering  the  analytic 
function  l//(z). 

§  3-32.   Elementary  treatment  of  Poissorfs  integral*.  To  find  the  limit 

lim  P  (ry  0) 

r->l 

it  will  be  assumed  in  the  first  place  that  /  (9)  is  integrable  according  to 

*  This  treatment  is  based  upon  that  given  in  Carslaw's  Fourier  Series  and  Integrals. 


240  Two-dimensional  Problems 

Ricmann's  definition  and  that  if  it  is  not  bounded  it  is  of  such  a  nature 
that  the  integral 

f(9')dff 

j  —it 

is  absolutely  convergent. 

Let  us  now  suppose  that  0  is  a  point  of  the  interval  —  TT  <  6  <  TT  which 
does  not  coincide  with  one  of  the  end  points.  We  shall  suppose  further 
that  the  limit 


,.      r  «  ,n  ,      x       «/n 
hm  [f(6+  r)  +  /(0- 


T->0 


/T^ 
......  (E) 


exists  and  is  equal  to  2F  (6),  where  F  (9)  is  simply  a  symbol  for  a  quantity 
which  is  defined  by  this  limit  when  6  is  chosen  in  advance.  No  knowledge 
of  the  properties  of  the  function  F  (9)  will  be  required. 

Now  let  a  function  <D  (#')  be  defined  for  all  values  of  6'  in  the  interval 
(—  TT  <  9'  <  TT)  by  the  equation 

<i>  (0')  =  f  (0')  -  F  (0). 
Then 


(0  -  9')[f(9')  -  F  (9)]  d9' 


P  (r,  9)  -  F  (9)  = 


Since,  by  hypothesis,  the  limit  (E)  exists,  a  positive  number  77  can  be 
found  so  as  to  satisfy  the  conditions 

\f(0  +  a)+f(0-a)-2F(0)\<€, 
when  0  <  a  <  17,     9  —  77  >  —  77,     9  +  77  <  77, 

e  being  an  arbitrary  small  positive  quantity  chosen  in  advance.  Then 


fo  +  y  fi 

K  (0  -  0')  O  (6')  dO'  =       K  (a)  [O  (0  +  a)  +  «»  (0  -  a)] 
fl-7)  JO 


da 


and  so 


(9)]  da, 


K(0-0')Q>  (O1) 


ri  TT 

<  €  \  K  (a)  da  <  e\      K  (a)  da  = 

JO  J-TT 

Also,  when  0  <  r  <  1, 

['  ~*/c  (0  -  9')  O  (fl')  dfl'  +  f  *    ic  (ff  -  0')  0  (0')  dfl7 

-rr  h  +  n 


277 


(77),  say, 
where  A  is  a  positive  quantity. 
But,  when  0<  r  <  1, 


(1 


-f  4r  sin2  77/2     2r  sin2  ij/2  ' 


Discussion  of  Poissorfs  Integral  241 

Hence  2irAK  (T?)  <  2-Tre  if  r  is  so  chosen  that 

Lz  r__    5 

2rsin2"7p<2' 
and  this  inequality  is  satisfied  if 

r>  s • 


Combining  the  two  results  we  find  that 

\P(r,B)-F  (9)  |  <  2e, 

if  !>,>[,  +  *  sin.*]'1. 

Hence  when  the  limit  (E)  exists, 

P(r,  0)  -*jF(0)     asr  ->  1. 

When  0  is  a  point  of  continuity  of  the  function/  (0)  we  have,  of  course, 
F  (0)  =  /  (0)  and  so  V  tends  to  the  assigned  value.  To  prove  that  V  is  a 
potential  function  when  r  <  1  it  is  sufficient  to  remark  that  the  series 


obtained  by  differentiating  (A)  term  by  term  with  respect  to  z  is  uniformly 

0  Y 

convergent  for  r<s<  1,  where  s  is  independent  of  r  and  0,  hence  -g-- 

12  T/ 

exists  and  is  a  function  of  z  only.  The  equation  x—  ~-  =  0  then  follows 

immediately. 

The  behaviour  of  Poisson's  integral  in  the  neighbourhood  of  a  point 
on  the  circle  at  which  /  (0)  is  discontinuous  is  quite  interesting.  Let  us 
suppose  that  /  (0)  has  different  values  f±  (0)  and  /2  (0)  when  the  point  0  is 
approached  along  the  circle  from  different  sides,  then  if  the  point  9  is 
approached  along  a  chord  in  a  direction  making  an  angle  air  with  the 
direction  of  the  curve  for  which  0  increases  the  definite  integral  tends  to 

the  value  ,  -  /m        f  /m 

(1  -«)A(0)  +  «/2(0). 

A  proof  of  this  theorem  is  given  by  W.  Gross,  Zeits.  f.  Math.  Bd.  n, 
S.  273  (1918).  When/  (0)  is  continuous  round  the  circle  we  have  the  result 
that  V  ->  /  (0)  as  any  point  on  the  circle  is  approached  along  an  arbitrary 
chord  through  the  point.  This  theorem  has  also  been  proved  by  P.  Pain- 
leve,  Comptes  Rendus,  t.  cxn,  p.  653  (1891)  and  by  L.  Lichtenstein,  Journ. 
f.  Math.  Bd.  CXL,  S.  100  (1911). 

EXAMPLES 
1     Show  by  means  of  Poisson's  formula  that  if 


-  1  (0  <  6  <  TT), 

16 


242  Two-dimensional  Problems 

the  potential  V  is  given  by  the  equation 

F  =  1  +  2  tan-1  ~  —  .—  .        (r2  <  a2). 
TT  2ar  sin  0  ' 

2.  Let  the  unit  circle  z  =  e*d  be  divided  into  n  arcs  by  points  of  division  Bt  ,  02  ,  .  .  .  0n  ,  where 
0  <  0X  <  62  <  ...  <  0n  =  2*.  Let  <f>  -f  i^  =  /  (z)  be  analytic  for  |  z  |  <  1  and  let  <£  satisfy  the 
following  conditions  on  the  circle 

*  =  <W     0m_l<6<0mt    c^-c^ 
cm  being  an  arbitrary  constant,  then 

27T/  (zj  -  -  27rCl  4-   2  (cm  -  c,)  [0,  +  2*  log  (e1^  -  z)]. 


[H.  Villat,  £W/.  rfe  la  Soc.  Math,  de  'France,  t.  xxxix,  p.  443  (1911);  "Aper9us  the"oriques 
sur  la  resistance  des  fluides,"  Scientia  (1920).] 

§  3-33.   Fourier  series  which  are  conjugate.  When  r  is  put  equal  to  1  in 
the  series  (A)  the  resulting  series  may  be  written  in  the  form 


n—  —oo 

and  is  the  "  Fourier  series"  associated  with  the  function/  (8). 

Separating  the  real  and  imaginary  parts,  the  series  may  be  written  in 
the  form  ^ 

a0  -f  S  (av  cos  v9  +  bv  sin  vd),  ......  (A') 


where  °*  =  J  (0>)  d0'  > 

av  =  -  [n  f(6')coav0'd6', 

TT  J  -IT 

6F=  L  [ 

T  J 


The  constants  a,,  6,,  are  called  the  "  Fourier  constants"  associated  with 
the  function/  (#').  In  terms  of  these  constants  the  series  for  V  is 

00 

x  7  =  a0-f  S  rv  (a,,  cos  v0  +  6^  sin  i/0).  ......  (B') 

v-l 

When  all  the  coefficients  are  real  the  series  for  the  conjugate  potential 

W  is 

60  -f-  r  (a±  sin  0  -  ^  cos  0)  +  r2  (a2  sin  0  -  62  cos  0)  +  ...  ,  ......  (C') 

and  this  is  associated  with  the  series 

60  -f  (ax  sin  0  —  &!  cos  0)  +  (a2  sin  0  —  b2  cos  0)  -f  ...,  ......  (D') 

which,  when  60  =  0,  is  called  the  conjugate*  of  the  Fourier  series  (A'). 
There  is  now  a  considerable  amount  of  knowledge  relating  to  the  con- 
jugate series.  One  question  of  importance  in  potential  theory  is  that  of  the 

*  Sometimes  it  is  this  series  with  the  sign  changed  which  is  called  the  conjugate  series.  See 
L.  Fejer,  Crelle,  vol.  CXLH,  p.  165  (1913);  G.  H.  Hardy  and  J.  E.  Littlewood,  Proc.  London  Math. 
Soc.  (2),  vol.  xxiv,  p.  211  (1926). 


Conjugate  Fourier  Series  243 

existence  of  a  function  g  (9)  of  which  the  foregoing  series  is  the  Fourier  series. 
In  this  connection  we  may  mention  a  theorem,  due  to  Fatou  *,  which  states 
that  if/  (6)  is  everywhere  continuous  and  the  potential  W  is  expressed  in  the 
form  W  —  W  (r,  0),  the  necessary  and  sufficient  condition  for  the  existence 
of  the  limit 

lim  W(r,0)  =  g(0)  ......  (E') 

r-+l 

for  any  assigned  value  of  6  is  that  the  limit 

lim    f  *  [/  (0  +T)  -/  (8  -r)]  cot  ldr  =  -  2^(0)      ......  (F) 

«->OJe  ^ 

should  exist.  Fatou  has  also  shown  that  if  /  (6)  has  a  finite  lower  bound 
and  is  such  that  /  (0)  is  integrable  in  the  sense  of  Lebesgue  then  the  limit 
(F')  exists  almost  everywhere. 

Lichtensteinf  has  recently  added  to  this  theorem  by  showing  that  the 
integral 

| 

J   —  7 

exists  when  ft/ 

J  —IT 

exists. 

For  further  properties  of  Poisson's  integral  and  conjugate  Fourier 
series  reference  may  be  made  to  the  book  of  G.  C.  Evans  on  the  logarithmic 
potential  J  and  to  Fichtenholz's  paper  in  Fundamenta  Mathematicae  (1929). 

Fatou's  expression  for  g  (9),  when/  (6)  is  given,  is 

9  (6)  =    277 


In  the  last  integral  the  symbol  P  denotes  that  the  integral  has  its  principal 
value.  Villat  has  deduced  this  expression  by  a  limiting  process  with  the  aid 
of  the  result  of  Ex.  2,  §  3-32.  The  formula  is  quite  useful  in  the  hydro- 
dynamical  theory  of  thin  aerofoils. 

An  alternative  expression,  obtained  by  an  integration  by  parts,  is 

g  (e]  =  L  f_/'  (f)  log  sin2 

§  3-34.  AbeVs  theorem  for  power  series.  When  for  any  fixed  value  of  0 
the  Fourier  series  converges  to  a  sum  which  may  be  denoted  for  the 
moment  by  g  (6),  it  may  be  shown  with  the  aid  of  a  property  of  power 
series  discovered  by  N.  H.  Abel  that  V  ->•  g  (6)  as  r  -*•  1.  But  since 

*  Acta  Math.  vol.  xxx,  p.  335  (1906). 

t  Crette's  Journ.  vol.  oxu,  p.  12  (1912). 

J  Amer.  Math.  Soc.  Colloquium  Publication*,  vol.  vi  (1927). 

16-2 


244  Two-dimensional  Problems 

V  ->  F  (0)  we  must  have  g  (9)  =  F  (0)  and  so  the  Fourier  series  represents 
F  (9)  whenever  it  is  convergent. 

The  series  for  V  may  be  written  in  the  form 

V  =  UQ  +  rul  +  r2uz+  ...,  (G') 

and  it  should  be  noted  that  the  coefficients  r,  r*, ...  occurring  in  the  different 
terms  are  all  positive  and  form  a  decreasing  sequence.  The  theorem  to  be 
proved  is  applicable  to  the  more  general  series 

V  =  t>o^o  +  fli^i  +  v2u2  + 
where  the  factors  v0,  vl9  v2  are  all  positive  and  such  that 

«Vu<  ^n,       v0=  !• 
Let  us  write 

«*0  =   ^0>        «1   =   ^0  +   %>        S2  =   U0  +  %  +  U2, 

and  suppose  that  the  quantities  s0 ,  ^ ,  s2 ,  ...  possess  an  upper  limit  H  and 
a  lower  limit  A,  then 

A  %  sn  <  //,         for  n  =  0,  1,  2,  .... 

Now  if  Vn  denotes  the  sum  of  the  first  n  +  \  terms  of  the  series  V, 
V  -  v0Uo  H-  ViUi  +  ...  ?;w?/n 

=    Oo  -    '^l)  ^0   +    (Vl  -    V2)  «!   +    ...    (Vn_!   ~    Vn)  5n_!   +   VnSn, 

and  in  this  series  not  one  of  the  partial  sums  sm  has  a  negative  coefficient. 
Hence 

Vn  <  K  -  vi)  7/  -f  (?;i  ~  va)  7/  -H  •••  K-i  -  ^n)  ^  +  vnH> 
and  Kn  >  (?'0  -  Vj)  li  +   (i\  -  v2)  h  +  ...  (vn-1  -  vw)  A  +   vwA. 

Summing  the  two  series  we  obtain  the  inequality 

v0A  -r  Fn  <  v0//, 

which  shows  that  |  Vn  \  <  vnk,  where  h  is  a  fixed  quantity  greater  than 
either  |  h  \  or  |  H  \ . 
Similarly,  if 

hnm  —  VmUm  +  vm+lum+l  ~f~   •••  vm+num+n> 

we  have  the  inequality 

I  7?  m  I  *f  11    Is 

I  •"'n      I  <•  vmKm> 

where  km  is  a  positive  quantity  greater  than  any  one  of  the  quantities 

I  um  | ,     |  nm  -f  um+l  | ,     |  um  +  um+1  +  ^m+2  | ,  .... 

If  now  the  series  UQ  4-  u^  4-  w2  +  •  •  •  ^s  convergent  and  « is  any  arbitrarily 
chosen  small  positive  quantity,  a  number  m  (e)  can  be  found  such  that 

|  Um  +  WWH-I+  ...  Um+n  |  <  €, 

for  n  ^  0,  1,  2,  ...  and  m  >  w  (f).  When  m  is  chosen  in  this  way  we  may 
take  km  =  €  and  since  vm  <  v0  <  1  we  have  the  inequality 

I  Rn™  I  <  €. 


AbeVs  Convergence  Theorem  245 

When  the  quantity  vn  is  a  function  of  a  variable  r  which  lies  in  the  unit 
interval  0  <  r  <  1  the  foregoing  inequality  shows  that  the  series  (G')  is 
uniformly  convergent  for  all  values  of  r  in  this  interval  and  so  represents 
a  continuous  function  of  r.  In  the  case  under  consideration  we  have 
vn  =  rn  and  the  conditions  imposed  on  vn  are  satisfied  if  0  <  r  <  1.  The 
function  V  is  consequently  continuous  at  r  —  1  and  so 

UQ  -f  u^  +  u2  +  ...  =  lim  P  (r,  0)  =  F  (9). 


i 


§3-41.  The  analytical  character  of  a  regular  logarithmic  potential*. 
Poisson's  integral  may  be  used  to  prove  that  a  logarithmic  potential  V 
which  is  regular  in  a  region  D  is  an  analytic  function  of  x  and  y. 

We  may,  without  loss  of  generality,  take  the  origin  at  an  arbitrary 
point  within  D.  Let  C  denote  the  circle  x2  -f  y2  —  a2  which  lies  entirely 
within  D,  then  for  a  point  x  =  a  cos  a,  y  —  a  sin  a  on  this  circle,  V  =  /  (a), 
where  /  (a)  is  a  continuous  function  of  a  and  so  by  Poisson's  formula 


V      "    J0  a2  f  r2-  2ar  cos  (0  -  a) ' 


where  x  =  r  cos  9,  y  =  r  sin  9  and  r  <  a. 
Now  the  series 


*.2  oo       / r\  n 

,— ,=  1  +  2   S   (')   cosn(0-«) 


a2  _|_  r2  __  2ar  cos  (9  —  a)  tl^i  \a) 

is  absolutely  and  uniformly  convergent  and  so  can  be  integrated  term  by 
term  after  being  multiplied  by  /  (a)  rfa/2?r.  Therefore 

F  =  a0  -f    2   f-  )    (an  cos  TI^  +  bn  sin  ?i0). 

n-l  \a/ 

Now  if  in  the  polynomial 

T\ 

-  )    (an  cos  ?i^  +  6n  sin  n9) 

a/ 

=  la~n  K  (»  +  *y)n  +  an  (#  -  *V)W   -  *n  (^  +  %)"  "I"  ^n  (#  ~  ^)W] 

we  replace  each  term  of  type  c,vqx*'tfl  ^y  ^s  modulus,  the  resulting  expres- 
sion will  be  less  than  the  corresponding  expression  obtained  by  doing  the 
same  thing  to  each  term  of  type  ePQxpyq  in  the  expansion  of  each  of  the 
four  binomials  and  adding  the  results.  Now  this  last  expression  is  less  than 


(T\ 
-  ) 
a/ 


where  M  is  the  upper  bound  of  an  and  bn  .   Now  let 

|  x  \  <  ,9,     |  y  |  <  s, 
where  s  <  a/2,  then 

2  [|  x  |  +  |  y  \]  Ma~n  <  2  (2s/a)n  M, 
and  the  series  of  moduli  is  convergent.  The  series  for  F  is  thus  a  power 

*  E.  Picard,  Cours  d*  Analyse,  t.  n,  p.  18. 


246  Two-dimensional  Problems 

series  in  x  and  y  which  is  absolutely  convergent  for  |  x  \  <  s,  \  y  \  <  s,  it 
thus  represents  an  analytic  function. 

Since,  moreover,  the  origin  was  chosen  at  an  arbitrary  point  in  D  it 
follows  that  V  is  analytic  at  each  point  within  D. 

For  the  parabolic    equation  ~-  =  y^  there  is  a  theorem  given  by 

Holmgren  *  which  indicates  that  z  is  an  analytic  function  of  x  in  the  neigh- 
bourhood of  a  point  (x0,  yQ)  in  a  region  R  within  which  z  is  regular. 

If  through  the  point  (x0,  yQ)  there  is  a  segment  a  <  y  <  b  of  the  line 
x  =  XQ  which  lies  entirely  within  J?,  there  is  a  number  c  such  that  for 
|  x  —  XQ  |  <  c,  a  <  y  <  b  there  is  an  expansion 

//r  _  T  ^2n  (rf  __ 

z  (x,  y)  =  S  eLL  #<•>  (y)  +  2 


where  $  (y)  =  *  (*o>  y),     $(y)  =  -faz  (xo>  V)- 

These  functions  <£  (?/),  0  (y)  are  continuous  (D,  oo)  in  a  <  y  <  6  and 
their  derivatives  satisfy  inequalities  of  type 

|  </>(n)  (y)  |  <  Mc~2n  (2n)  !,     |  0(n)  (y)  |  <  Mc~*n  (2n)  !  . 


§  3-42.    Harnack's  theorem^.   Let  JF  a  ,  for  each  positive  integral  value  of 
,9,  be  a  potential  function  which  is  continuous  (D,  2)  (i.e.  regular)  in  a  closed 
region  R  and  let  the  infinite  series 

^i  4-  w2  +  w3  +  ...  ......  (A) 

converge  uniformly  on  the  boundary  J5  of  R,  then  the  series  converges 
uniformly  throughout  R  and  represents  a  potential  function  which  is 
regular  and  analytic  in  R.  The  sum  wn  -f  ivn+l  +  ...  wn+v  ig  a  potential  func- 
tion regular  in  R.  If  it  is  not  a  constant  it  assumes  its  extreme  value  on  B 
and  if  Np  is  the  numerically  greatest  of  these  we  shall  have 

|  wn  +  wn+l+  ...  wn+P  I  <  \N9\. 

Since,  however,  the  series  converges  uniformly  on  B  we  can  choose  a  num- 
ber m  (e)  such  that  when  n  >  m  (c)  we  have 

\N,\<*. 

for  all  positive  integral  values  of  p.  This  inequality,  combined  with  the 
previous  one,  proves  that  the  series  (A)  converges  uniformly  in  R  and  so 
represents  a  continuous  function  w.  Now  let  C  be  any  circle  which  lies 
entirely  within  7?  and  let  Poisson's  formula  be  used  to  obtain  expressions 
for  potential  functions  W,  Wly  W2,  W3,  ...  regular  within  C  and  having 
respectively  the  same  boundary  values  as  the  functions  W9wl9w2,w3,  ... 

*  E.  Holmgren,  ArJnv  for  Mat.,  Astr.  och  Fyaik,  Bd.  I  (1904);  Bd.  m  (1906);  Bd.  iv  (1907); 
Comptes  Rendus,  t.  CXLV,  p.  1401  (1907). 

t  Kellogg  calls  this  Harnack's  first  theorem.  See  Potential  Theory,  p.  248.  The  theorem  was 
given  by  Harnack  in  his  book.  It  has  been  extended  to  other  equations  of  elliptic  type  by 
L.  Lichtenstein,  Crelle'a  Journal,  Bd.  CXLII,  S.  1  (1913). 


Analytical  Character  of  Potentials  247 

Since  a  potential  function  with  assigned  boundary  values  on  C  is  unique 
if  it  is  required  also  to  be  regular  within  C  we  have  Ws  —  ws  (s  =  1,  2,  ...). 
Furthermore,  since  the  series  (A)  is  uniformly  convergent  it  may  be  inte- 
grated term  by  term  after  multiplication  by  the  appropriate  Poisson  factor. 
Therefore  at  any  point  within  £ 

w  =  w,+  w2+  w3+  ... 

=   U\  ~\~   W2   -f   M>3  +    ...    =  W. 

Hence  within  C  the  function  w  is  identical  with  the  regular  potential  func- 
tion which  has  the  same  values  as  w  at  points  on  C.  Since  C  is  an  arbitrary 
circle  within  R  it  follows  that  w  is  a  regular  potential  function  at  all  points 
of  R  and  is  consequently  analytic  at  each  point  of  R. 

For  recent  work  relating  to  the  analytical  character  of  the  solutions  of 
elliptic  partial  differential  equations  reference  may  be  made  to  L.  Lichten- 
steiri,  Enzyklo'padie  der  Math.  Wiss.,  n  C.  12 ;  T.  Rado,  Math.  Zeits.  Bd.  xxv, 
S.  514  (1926);  S.  Bernstein,  ibid.  Bd.  xxvm,  S.  330  (1928);  H.  Lewy, 
Gott.  Nachr.  (1927),  Math.  Ann.  Bd.  ci,  S.  609  (1929). 

§  3-51.  Scliwarz's  alternating  process.  H.  A.  Schwarz*  has  used  an  alter- 
nating process,  somewhat  similar  to  that  used  by  R.  Murphy  f  in  the  treat- 
ment of  the  electrical  problem  of  two  conducting  spheres,  to  solve  the  first 
boundary  problem  of  potential  theory  for  the  case  of  a  region  bounded  by 
a  contour  made  up  of  a  finite  number  of  analytic  arcs  meeting  at  angles 
different  from  zero. 

To  indicate  the  process  we  consider  the  simple  case  of  two  contours 
aa,  6/3  bounding  two  areas  A,  B  which  have  a  common  part  C  bounded 
by  a  and  /?,  while  a  and  6  bound  a  region  D  represented  by  A  f  B  —  C. 

We  shall  use  the  symbols  a,  by  a,  /j  to  denote  also  the  parameters  by 
means  of  which  the  points  on  these  curves  may  be  expressed  in  a  uniform 
continuous  manner  and  shall  use  the  symbols  m  and  n  to  denote  the  points 
common  to  the  curves  a  and  b.  We  shall  suppose,  moreover,  that  the 
choice  of  parameters  is  made  in  such  a  way  that  in  and  n  are  represented 
by  the  parameters  m  and  n  whether  they  are  regarded  as  points  on  a,  6,  a 
or  /?.  This  can  always  be  done  by  subjecting  parameters  chosen  for  each 
curve  to  suitable  linear  transformations. 

Our  problem  now  is  to  find  a  potential  function  V  which  is  regular 
within  D  and  which  satisfies  the  boundary  conditions  V  —  f  (a)  on  a, 
V  -  g  (b)  on  6,  where  /  (m)  =  g  (m),  f  (n)  =  g  (n). 

•     We  shall  suppose  that  /  (a)  is  continuous  on  a  and  that  g  (b)  is  con- 
tinuous on  b.    We  shall  suppose  also  that  a  function  h  (a),  which  is  con- 
tinuous on  a,  is  chosen  so  as  to  satisfy  the  conditions 
h(m)  =/(m),     h  (n)  =/(/&). 

*  Berlin  MonatsberichU  (1870);  Gesammelte  Werke,  Bd.  n,  S.  m 
t  Electricity,  p.  93,  Cambridge  (1833). 


248  Two-dimensional  Problems 

We  now  form  a  sequence  of  logarithmic  potentials  u1}u2)  ...  regular  in 
A,  and  a  sequence  of  logarithmic  potentials  vl9  v2,  ...  regular  in  J3;  these 
potentials  being  chosen  so  as  to  satisfy  the  following  boundary  conditions 
in  which  us  (/?)  denotes  the  value  of  us  on  /3,  and  vs  (a)  denotes  the  value  of 

vs  on  a,  (s  ~  1,  2,  ...): 


on  a,  u^  —  ti  (a)  on  a, 
u2  =  /  (&)  on  a,  i£2  =  v±  (a)  on  a, 
u3  ^  f  (a)  on  a,  ^3  =  v2  (a)  on  a, 

^  =  gr  (ft)  on  6,  #!_  =  Uj  (j8)  on  /3, 
^2  ~  0  CO  on  6,  v%  —  u2  (/3)  on  /3, 
#g  -~  r/  (6)  on  6,  ?>3  —  ?/3  (/3)  on  /3, 

Writing      ws.  —  fuL  -f-  (?/2  —  ?^j)  |-  (w3  —  w2)  -h  ...  (uh  —  us^i), 

our  object  now  is  to  show  that  as  .9  ->  oo  the  series  for  us  and  vs  converge 
and  represent  potentials  which  are  exactly  the  same  in  C.  To  establish  the 
convergence  of  the  series  we  shall  make  use  of  the  following  lemma. 

We  note  that  ws  =  us  —  11^^  is  a  logarithmic  potential  which  is  regular 
in  A  and  which  is  zero  on  a.  Let  S,.  (a)  be  its  value  at  a  point  on  a  and  let 
83  be  the  maximum  value  of  |  8S  (a)  \ . 

Now  let  </>  be  the  logarithmic  potential  which  is  regular  in  A  and  which 
satisfies  the  boundary  conditions  (/»  -=  1  on  a,  0  —  0  on  a.  As  the  point 
(x,  y)  approaches  one  of  the  points  of  discontinuity  m,  <f>  tends  to  a  value 
9  such  that  0  <  6  <  1.  Now  a  regular  potential  function  attains  its  greatest 
value  in  a  region  on  the  boundary  of  the  region,  therefore  <f>  <  0  for  all 
points  of  A  and  so  there  is  a  positive  number  e  between  0  and  1  such  that, 
on  /?,  <f>  <  e  <  1 . 

Now  ws  -f  Ss,<£  is  zero  on  a  and  positive  on  a  and  is  a  logarithmic 
potential  regular  in  A .  Its  least  value  is  therefore  attained  on  the  boundary 
of  A  and  so  ws  f  £<.(/>  >  0  within  A.  This  inequality  may  be  written  in  the 

f°rm  o/i         \   ,          .     *       n 

os  (<p  —  €j  ~f-  w\  -!-  to,  >  0, 

and  since  </>  <  e  on  /3  it  follows  that  ws  4  e8s  >  0  on  j8. 

In  a  similar  way  we  can  show  that  ivs  —  8s<f>  <  0  in  A  and  so  we  may 
conclude  that  ws  —  e8s  <  0  on  /?.  Combining  the  inequalities  we  may  write 

The  number  e  was  derived  from  the  function  <f>  associated  with  A.  In  a 
similar  way  there  is  a  number  T?  associated  with  the  region  B  and  the 
curve  a.  Let  K  be  the  greater  of  these  two  numbers  if  the  two  numbers  are 
not  equal. 


Schwarz*s  Alternating  Process  249 

Writing  ts  =  vs  ~  vs^  and  using  the  symbol  rs  (/3)  to  denote  the  value 
of  ts  on  p  we  use  ra  to  denote  the  maximum  value  of  |  rs  (/3)  |  on  /?.  We  then 
find  in  a  similar  way  that  ,  M  , 

K  I  <  TS, 

and  so  we  may  write 

|    Wa    |    <    K8S1         |    t,        <    KTS. 

We  thus  obtain  the  successive  inequalities 

I  «'2  ~  *'i  I  '-"  I 
Therefore  r2 


\     on  «. 

<j         -  -         j  i 

Therefore  S^  <  /cr2  <  *2S2,  ...     S,+2  <  *2sS2, 

T3  <    *S3   <    K2T2,    ...         Ts+2  <    *2*T2. 

The  series  for  ?/,  and  vs  thus  converge  uniformly  at  all  points  of  the 
boundary  of  C  and  so  by  Harnack's  theorem  represent  regular  logarithmic 
potentials  which  we  may  denote  by  ?/  and  v  respectively.  Since  ?/,,  -  v^^ 
on  a  and  us  =  vs  on  /3  it  follows  that  u  -=  v  on  the  boundary  of  6"  and  so 
u  =  v  throughout  C.  Since,  moreover,  the  series  for  u  converges  uniformly 
on  the  boundary  of  A  and  the  series  for  v  converges  uniformly  on  the 
boundary  of  B  these  series  may  be  used  to  continue  the  potential  function 
u  —  v  beyond  the  boundary  of  G  into  the  regions  A  and  B,  and  the  potential 
function  thus  defined  will  have  the  desired  values  on  a  and  6. 

§  3-61.  Flow  round  a  circular  cylinder.  To  illustrate  the  use  of  the  com- 
plex potential  in  hydrodynamics  we  shall  consider  the  flow  represented  by 
a  complex  potential  %  which  is  the  sum  of  a  number  of  terms 

y    ——    "* 

Xl  -  U  (z  +  a2/z),     X2  =  M  log  z,     ^  =  ™  log         ~°  , 

z  --  zt 

•   ,  i         2  --  22 
X4=   -1C    log  2. 

^          ^3 

Writing  z  =  re^,  ^  =  ^>  4-  i^  we  consider  first  the  case  in  which  x  =  Xi 
and  £7  is  real.  We  then  have 

u~-  iv=  dx/dz  =  T7  (  1  -  a2/z2)  , 

^  =  C7  sin  0  (r  -  a2/^)- 

The  stream-function  </r  is  zero  on  the  circle  r2  =  a2  and  also  on  the  line 
y  =  0.  There  is  thus  one  stream-line  which  divides  into  two  parts  at  a 
point  S  where  it  meets  the  circle  ;  these  two  portions  reunite  at  a  second 
point  Sf  on  the  circle  and  the  stream  -line  leaves  the  circle  along  the 
line  y  =  0.  Since  z2  =  a2  at  the  points  S  and  8'  these  points  are  points  of 
stagnation  (u  =  v  =  0).  It  will  be  noticed  that  the  stream-line  y  =  0  cuts 
the  boundary  r2  =  a2  orthogonally.  This  is  in  accordance  with  the  general 
theorem  of  §  1-72. 


250  Two-dimensional  Problems 

At  a  great  distance  from  the  circle  we  have  u  —  iv  =  U,  fy  =  C/y,  and 
so  the  stream-lines  are  approximately  straight  lines  parallel  to  the  axis 
of  x.  Our  function  i/j  is  thus  the  stream  -function  for  a  type  of  steady  flow 
past  a  circular  cylinder.  This  flow  is  not  actually  possible  in  nature,  the 
observed  flow  being  more  or  less  turbulent  while  for  a  certain  range  of 
speed  depending  upon  the  viscosity  of  the  fluid  and  the  size  of  the  cylinder, 
eddies  form  behind  the  cylinder  and  escape  downstream  periodically*  in 
such  a  way  as  to  form  a  vortex  street  in  which  a  vortex  of  one  sign  is 
almost  equidistant  from  two  successive  vortices  of  the  opposite  sign  and 
each  vortex  of  this  sign  is  almost  equidistant  from  two  successive  vortices 
of  the  other  sign.  Vortices  of  one  sign  lie  approximately  on  a  line  parallel 
to  the  axis  of  x  and  vortices  of  the  other  sign  on  a  parallel  line. 

Some  light  on  the  formation  of  this  asymmetric  arrangement  of 
vortices  is  furnished  by  a  study  of  the  equilibrium  and  stability  of  a  pair 
of  vortices  of  opposite  signs  which  happen  to  be  present  in  the  flow  round 
the  circular  cylinder. 

The  flow  may  be  represented  approximately  by  writing 

X  =  Xi  +  X*  +  *4> 

and  choosing  20,  zl9  z2,  z3  so  that  the  circle  r2  =  a2  is  a  stream  -line,  This 
condition  may  be  satisfied  by  writing 


ItA,B,C,D  are  the  points  specified  by  the  complex  numbers  z0  ,  zl  ,  z2  ,  z3  , 
respectively,  these  equations  mean  that  B  is  the  inverse  of  A  and  0  the 
inverse  of  D. 

In  the  theory  of  Helmholtz  and  Kelvin  vortices  move  with  the  fluid. 
When  the  vortices  are  isolated  line  vortices  this  result  is  generally  replaced 
by  the  hypothesis  that  the  velocity  of  any  rectilinear  vortex  to  is  equal 
to  the  resultant  of  the  velocities  produced  at  its  location  by  all  the  other 
vortices  which  together  with  o>  produce  the  resultant  flow  at  an  arbitrary 
point.  In  using  this  hypothesis  the  uniform  flow  U  is  supposed  to  be 
produced  by  a  double  vortex  at  infinity  and  the  complex  potential  Ua2/z 
is  interpreted  as  that  of  a  double  vortex  at  the  origin  of  co-ordinates  0. 

The  vortex  at  A  will  be  stationary  when 


Taking  for  simplicity  the  cavse  when  r2  =  r0,  ft2  =  —  00,  c'  =  c,  and 

*  Th.  v.  lUrmAn,  Odtt.  Nachr.  p.  547  (1912);  PJiys.  Zeits.  p.  13  (1912).  The  vortices  have 
been  observed  experimentally  by  Mallock,  Proc.  Roy.  Soc.  London,  vol.  ix,  p.  262  (1907);  and 
by  Benard,  Comptes  Rendus,  vol.  CXLVJI,  pp.  839-970  (1908);  vol.  CLVI,  pp.  1003-1225  (1913); 
vol.  OLXXXII,  pp.  1375-1823  (1926);  vol.  CLXXXIII,  pp.  20-184  (1920). 


Cylinder  and  Isolated   Vortices  251 

separating  the  real  and  imaginary  parts  of  the  expression  on  the  right, 
after  multiplying  it  by  20  ,  we  obtain  the  equations 

0  =  U  (r0  -  a2rQ~l)  cos  00  -  ±c  cot  0Q  +  a*crQ2£l-1  sin  200  , 
0  =  U  (r0  +  a2r0-!)  sin  00  -  cr02/(r02  -  a2)  -  \c  +  cr02  (r02  -  a2  cos  200)  Q-i, 
where  £3  =  r04  -  2a2r02  cos  200  +  a4. 

The  first  equation  gives 

2UQ,  sin  00  =-  cr0  (a2  -  r02), 
and  when  this  value  of  U  is  substituted  in  the  second  equation  it  is  found 

i/nat)  9        «        i    ci   9    •     /i 

r02-  a2  =  ±  2r02sm00. 

This  result  was  obtained  by  Foppl*,  who  also  studied  the  stability  of  the 
vortices.  The  result  tells  us  that  the  vortex  can  be  in  equilibrium  if 
AB  =  AD.  To  confirm  this  result  by  geometrical  reasoning  we  complete 
the  parallelogram  BADE  and  determine  a  point  N  on  the  axis  of  y  such 
that  ON  =  AN.  Let  M  be  the  point  of  intersection  of  BC  and  AN  ,  G  the 
point  of  intersection  of  AC  and  BD. 

On  the  understanding  that  all  lines  used  to  represent  velocities  are  to 
be  turned  through  a  right  angle  in  the  clockwise  direction  the  velocities 
at  A  due  to  the  different  vortices  may  be  represented  as  follows  : 

Those  due  to  the  vortices  at  B  and  D  by  c/AB  and  c/AD  respectively. 
Since  AB  =  AD  these  two  velocities  together  may  be  represented  by 
c.AE/AB*  along  AE. 

The  velocity  due  to  the  vortex  at  G  may  be  represented  by  c/CA  along 
GA  and  equally  well  by  cGA/AB*  along  GA.  The  resultant  velocity  at  A 
due  to  the  vortices  at  B,  C  and  D  may  thus  be  represented  by  c.GE/AB2 
along  OE. 

On  the  other  hand,  the  velocity  U  is  represented  by  U  along  ON,  and 
the  velocity  due  to  the  double  vortex  at  O  by  U  .NM/ON  along  NM.  The 
velocity  in  the  flow  round  the  cylinder  in  the  absence  of  the  vortices  is 
thus  represented  by  U  .OMJON. 

A  A  A 

Now  MAE  =  ME  A  =  OCM,  therefore  0,  M  ,  A,  C  are  concyclic  and  so 
0&LC  =  OAC  -  OEG.  This  means  that  OM  and  EG  are  parallel.  By 
choosing  c  so  that  c.EG/AB2=  U.OM/ON  the  resultant  velocity  at  A 
will  H  zero.  Since  the  triangles  ON  A,  OAD  are  similar,  the  equation  for 
c  becomes  simply 


T7  OM  AB2      T7  OM  AB*      TJ  OM  A  n       A  ^  TT  , 
c  -  U  ON  EG  -  U  ON  AG  =  U  ON  A  °  =  A  C*  '  U'a 

-  U  (r02  -  a2)  (1  -  a4/r04)/a 

and  implies  that  the  strength  of  the  vortex  at  A  is  greater  the  greater  the 
distance  of  A  from  the  origin. 

*  L.  Foppl,  Munchen  Sitzungsber.  (1913).  See  also  Howland,  Journ.  Roy.  Aeron.  Soc.  (1925); 
M.  Dupont,  La  Technique  Atronautique,  Dec.  15  (1926)  and  Jan.  15  (1927);  W.  G.  Bickley,  Proc. 
Roy.  Soc.  Lond.  A,  vol.  cxix,  p.  146  (1928). 


252  Two-dimensional  Problems 

The  stream-lines  in  the  flow  studied  by  Foppl  are  quite  interesting  and 
have  been  carefully  drawn  by  W.  Miiller*.  There  are  four  points  of 
stagnation  on  the  circle,  two  of  these,  8  and  8' ',  lie  on  the  line  y  =  0,  while 
the  other  two,  S0,  $</,  are  images  of  each  other  in  the  line  y  =  0.  Stream- 
lines orthogonal  to  the  circle  start  at  SQ  and  S0'  and  unite  at  a  point  T  on 
the  line  y  where  they  cut  this  line  orthogonally.  This  point  T  is  also  a 
point  of  stagnation.  Outside  these  stream -lines  the  flow  is  very  similar 
to  that  round  a  contour  formed  from  arcs  of  two  circles  which  cut  one 
another  orthogonally;  within  the  region  bounded  by  these  stream-lines 
there  is  a  circulation  of  fluid  and  the  flow  between  T  and  the  circle  is 
opposite  in  direction  to  that  of  the  main  stream.  The  stream-lines  are, 
indeed,  very  similar  to  those  which  have  been  frequently  observed  or 
photographed  in  the  case  of  the  slow  motion  round  a  cylinderf. 

Let  us  now  consider  the  case  when  there  is  only  one  vortex  outside  the 
cylinder  and  a  circulation  round  the  cylinder.  We  now  put 

A         Al    '    A2    '    A3  * 

In  this  case 

u  -  w  ={7(1-  a*/z2)  -f  ik/z  -f  ic  [(z  -  rQetd»)~l  -  (z  -  v^aVo)-1], 
and  the  component  velocities  of  the  vortex  A  are  given  by 

while  for  its  image  B 

?/!  -f  ii\  -=  —  a2  (UQ  —  ivQ)  r0~2e2lV 

If  X,  Y  are  the  components  of  the  resultant  force  on  the  cylinder  per 
unit  length,  we  have 

X  +  iY  --=  -  Jpa  P*  (w2  +  v2  -f-  2  -£}  e"dO  -  (Xg  +  iYg)  -I-  (X*  +  iY+),  say. 
J  o    V  v*  / 

Now  when  z  —  aet09 
u"  +  v2  =  4C/2  sin2  8  +  k*fa2  4-  c2  (r02  -  a2)2/a27?4 

-I-  4C7  sin  0.[*/a  -  c  (r02  -  a2)/a.R2J  -  2kc  (r02  -  a2)/a2J?2, 
where  7^2  =  a2  +  r02  -  2ar0  cos  (/9  -  00). 

Therefore  ^  +  iYq  =  2-rrip  {kU  —  c  (UQ  -f-  iv0)}. 

We  have  also  for  r  —  a 

*  Znts.fur  techmsche  Physik,  Bd.  vm,  S.  62  (1927);  Mathematische  Stromungslehre  (Springer, 
Berlin,  1928),  p.  124. 

t  See  especially  the  photographs  published  by  Camichei  in  La  Technique  A  fronautique,  Nov.  15 
(1925)  and  Dec.  15(1925). 


Circulation  round  a  Cylinder  and  Vortex  253 

therefore 

\*'W  , .  .-9f| 


n 


*  a/ 

a<£         .  9</r 

a*  +  *  dt 

2rr   gj, 

« -  aet<?  <i#  ==  Trie  (Uj  +  ivj)  ~  7rica2rQ~2  (UQ  —  iv0)  e2*eo  —  2Tric  (t^  -f  ivj). 
o    ot 

Combining  these  results  we  have 

X  +  iY  =  27rip  {/J?7  —  c  (w0  +  i#0)  -f  c  (^  +  i#i)}. 

This  result  may  be  extended  to  the  case  in  which  there  are  any  number 
of  vortices  outside  the  cylinder*,  the  general  result  being 

-X"  -f  iY  =  2  Trip  \kU  —  2  cs  (?/2s  4-  iv2s  — 

In  the  special  case  when  there  is  only  one  vortex  and  k  =  0,  00  =  0, 
we  have  ut  -f  iz^  =  —  a2rQ~2  (UQ  ~  ivQ), 

„,  MJ     TT  /I  n2r  — 2\          ^>r     /r  2          /t2\-l 

U-Q frt/Q \J        ^1      U/       f  Q  f      t/Ll  Q      \'   Q          ^       /  > 

Z  f  iF  =  27rpc  [c/rQ  —  iU  (I  ~  a4r0~4)]. 

Introducing  the  coefficients  of  lift  and  drag,  defined  by  X  ~  p8U2.CJ)9 
Y  =  pSU2.CL,  8  being  the  projected  area,  we  find 

CD  =  (c/aU)2  7ra/r0,     CL  =  -  TT  (c/aC7)  (1  -  a*r0-4). 

These  results  were  obtained  by  W.  G.  Bickleyf  who  plots  the  lift-drag 
curves  for  r  -  2a,  4a  and  6a,  and  compares  them  with  the  published 
curves  for  Flettner  rotors  (rotating  cylinders  with  end  plates).  With  the 
last  two  values  the  agreement  is  fair  except  for  low  values  of  the  lift. 

The  stream-lines  for  the  case  of  a  single  vortex  outside  the  cylinder 
have  been  drawn  by  W.  MiillerJ. 

EXAMPLES 

1.  If  in  a  type  of  flow  similar  to  that  considered  by  Foppl  the  vortices  at  20  and  z2  are  not 
images  of  each  other  in  the  line  y  =  0,  one  of  the  conditions  that  the  vortices  may  be  stationary 
in  the  flow  round  the  cylinder  is 

(r0  -  a2rQ~l)  cos  00  -  (r2  -  a*r2~l)  cos  0a. 

2.  If  in  Foppl's  flow  the  vortices  move  so  that  they  are  always  images  of  each  other  in 
the  line  y  —  0  the  resultant  force  on  the  cylinder  is  a  drag  if 

4r0*  sin2  00  >  (r02  -  a2)2. 

[Bickley.] 

*  H.  Bateraan,  Bull.  Amer.  Math.  Soc.  vol.  xxv,  p.  358  (1919);  D.  Riabouchinsky,  Comples 
Rendus,  t.  CLXXV,  p.  442  (1922);  M.  Lagally,  Zeits.  f.  angew.  Math.  u.  Mech.  Bd.  n,  8.  409  (1922). 
In  this  formula  the  even  suffixes  refer  to  the  vortices  outside  the  cylinder  and  the  odd  suffixes 
to  the  image  vortices  inside  the  cylinder. 

t  Loc.  cit.  ante,  p.  251. 

t  Loc.  cit.  ante,  p.  252. 


254  Two-dimensional  Problems 

3.  A  plate  of  width  2a  is  placed  normal  to  a  steady  stream  of  velocity  U  and  vortices 
form  behind  the  plate  at  the  points 

Prove  that  the  conditions  are  satisfied  by 


=  *. 

8  (*»  +  «*)*  -(*•»  +  «')* 
Prove  also  that  when 


-  *0  +  2  (  V  +  a2)*, 

the  velocity  does  not  take  infinite  values  at  the  edges  of  the  plate  and  the  vortices  are 
stationary. 

[D.  Riabouchinsky.] 

§  3-71.  Elliptic  co-ordinates.  Problems  relating  to  an  ellipse  or  an 
elliptic  cylinder  may  be  conveniently  solved  with  the  aicf  of  the  sub- 

stitution .     .  U  /<"    ,     •    \  1     Y 

x  +  ly  =  c  cosh  (f  +  2,7?)  =  c  cosh  f  , 

which  gives 

x  =  c  cosh  £  cos  77, 

T/  =  c  sinh  £  sin  77. 
The  curves  £  =  constant  are  confocal  ellipses, 

52        +     j£__  =  l 
c2  cosh2  f     c2  sinh2  £         5 

the  semi-axes  of  the  typical  ellipse  being  a  =  c  cosh  ^  and  6  =  c  sinh  £. 
The  angle  77  can  be  regarded  as  the  excentric  angle  of  a  point  on  the 
ellipse. 

The  curves  77  =  constant  are  confocal  hyperbolas,  the  semi-axes  of  the 
typical  hyperbola  being  a'  —  c  cos  77  and  6'  =  c  sin  77. 

The  first  problem  we  shall  consider  is  that  of  the  determination  of  the 
viscous  drag  on  a  long  elliptic  cylinder  which  moves  parallel  to  its  length 
through  the  fluid  in  a  wide  tube  whose  internal  surface  is  a  confocal 
elliptic  cylinder*. 

Considering  a  cylindrical  element  of  fluid  bounded  by  planes  parallel 
to  the  plane  of  xy  and  a  curved  surface  generated  by  lines  perpendicular 
to  this  plane,  the  viscous  drag  per  unit  length  on  the  curved  surface  of  the 
cylinder  is 


taken  round  the  contour  of  the  cross-section,  w  being  the  velocity  parallel 
to  a  generator  and  p  being  the  coefficient  of  viscosity. 

If  the  fluid  is  not  being  forced  through  the  tube  under  pressure  the 
pressure  may  be  assumed  to  be  constant  along  the  tube  and  so  in  steady. 
motion  the  total  viscous  drag  on  the  cylindrical  element  must  be  zero. 

*  C.  H.  Lees,  Proc.  Roy.  Soc.  A,  vol.  xcn,  p.  144  (1916). 


Viscous  Resistance  to  Towing  255 

Transforming  the  line  integral  into  an  integral  over  the  enclosed  area,  we 

obtain  the  equation 

d*w     Shu 

fo2  +  ay2  ~    ' 

The  boundary  conditions  are  w  =  0  when  £  =  ^  ,  and  w  =  v  when 
£  =  £2  »  we  therefore  write 


Since  T?  varies  from  0  to  2-rr  in  a  complete  circuit  round  the  contour  of 
the  cross-section,  the  total  viscous  force  per  unit  length  of  the  cylinder  is 

27TfJLV    __  2-JT^JLV 

S^=r^2  ~  log  (aT+  b~]  T~Togr(a2  "+  62)  ' 

If  the  inner  ellipse  reduces  to  a  straight  line  of  length  2c,  the  total  drag 
•  on  the  plane  is  D  per  unit  length,  where 

D  [log  (a,  +  6,)  -  log  (2c)]  =  2^, 
and  the  resistance  per  unit  area  at  the  point  x  is 

(D/27r)  (c2-  z2)-*. 

It  is  clear  from  this  expression  that  the  resistance  per  unit  area,  i.e.  the 
shearing  stress,  is  much  greater  near  the  edges  of  the  strip  than  near  its 
centre  line. 

The  foregoing  analysis  may  be  used  with  a  slight  modification  to 
determine  the  natural  charges  on  two  confocal  elliptic  cylinders  regarded 
as  conductors  at  different  potentials.  If  V  is  the  potential  at  (f,  77)  and 
V  =  0  for  £  =  £x  ,  V  =  v  for  f  =  £2  ,  we  have 

y  &  -  &)  =  »  (&  -  a 

and  the  density  of  charge  on  the  cylinder  £  =  ^  is 

i  aF3£     i 


_ 

~  477  8£  3/1  -~  47T  87  35  ^  ST(^  -  &)  c 

When  the  inner  cylinder  reduces  to  the  strip  whose  cross-section  is 
S^^  we  have,  when  v  =  2  (£x  —  ^2), 

CTj  =    (1/277-C)  COS6C  T^u 

and  if,  moreover,  the  outer  cylinder  is  of  infinite  size  ol  becomes  the  natural 
charge  on  the  strip  when  the  total  charge  per  unit  length  is  equal  to 
unity;  this  is  the  charge  density  on  each  side  of  the  strip. 

To  find  the  stream-function  for  steady  irrotational  flow  round  an 
elliptic  cylinder  when  there  is  no  circulation  round  the  cylinder,  we  write 
^  =  ^i  -f-  02  >  where 

^i  ^  Uy  ~  Vx  =  c  (U  sinh  ^  sin  77  —  V  cosh  £  cos  77) 
is  the  stream-function.  for  the  steady  flow  at  a  great  distance  from  the 


256  Two-dimensional  Problems 

cylinder  and  ^2  is  the  stream-function  for  a  superposed  disturbance  in  this 
flow  produced  by  the  cylinder.  To  satisfy  the  boundary  condition  </>  =  0 
at  the  surface  of  the  cylinder,  and  the  condition  that  the  component 
velocities  derived  from  </f2  are  negligible  at  infinity,  we  write 

02  =  e~*  (A  cos  ?;  -f  B  sin  ??). 

Choosing  the  constants  so  that  tft  —  0  on  the  cylinder,  we  have 
e-fi  A  =  cV  cosh  &  ,     e~fi  B  =  —  cU  sinh  & 

=  ^7  =-6tZ7 

where  al5  6X  are  the  semi-axes  of  the  ellipse  £  =  &.   We  have  also 

«!-{-&!  =  c^-^i,     ax  —  fej_  =  Cjg-fi 
Therefore    02  =  (ax  +  6^*  (%  —  6^"*  e~f  (7%  cos  T?  —  f/fij  sin  TJ), 

<f>  +  i*fi=  (U  -  iV)  z  -f  (*//>!  +  iF«j)  (aj  +  6^4  (%  -  &!)-*  e-f. 

To  find  the  electrical  potential  of  a  conducting  elliptic  cylinder  which 
is  under  the  influence  of  a  line  charge  parallel  to  its  generators,  we  need  an 
expression  for  the  logarithm 

log  (z0  -  z)  =--  log  [c  (cosh  £0  -  cosh  £}] 

-  £0  -{-  log  Jc  +  log  (1  -  e^-^o)  (l  -  e~^-^o) 

-  &  +  logic  -2  S  w^e- 
Writing  this  equal  to  (/>0  -f  ifa  we  have 

00 

^o  "-^  ^o  -H  l°g  ^°  —  2  S  n-1e~nfo  (cosh  nf  cos  nrj  cos 
>i-i 

+  sinh  ng  sin  ?^r;  v^in  nrjQ). 

To  obtain  a  potential  which  is  constant  over  the  elliptic  cylinder 
f  ^  f  i  >  wo  write  </>  ==  ^0  4-  ^  ,  where 

00 

^>1  —    £  (^rlne~"£  cos  Tir;  -f  Bne~^  sin  w^). 

71-1 

Each  term  of  this  series  is  indeed  a  potential  function  which  vanishes 
at  infinity.  Choosing  the  constants  An  ,  Bn  ,  so  that  the  boundary  condition 
<£  --=  0  on  £  =  ^  is  satisfied  by  <f>  =  <f>0  +  <f>l9  we  have 
nAne~n%i  =  2e~n^o  cosh  T^  cos  TIT^Q, 
nBne~nti  =  2e~n^o  sinh  n^  sin  n^. 
Hence  when  ^  <  ^  <  £0, 


^  ~  f  o  +  ^°g  lc  +  2  7i-1en(^i"^o>  sinh  TZ-  (^  —  g) 

n~  I 

Summing  the  series  we  find  that 

.-.«-J»  eL 


Induced  Charge  Density  257 

The  corresponding  stream-function  is 


*  -  ,  +  tan-  -_  IL     —.+  tan- 

r       '  1  —  e*-fo  cos  (770  —  ??) 

and  when  ^  =  0  the  value  of  —  for  £  =  0  is 


__    _         __ 

cosh  £0  -  cos  (770  -  77)  ' 
The  surface  density  of  the  charge  on  the  plate  £  =  0  is  thus 

1   dri  [i  _    ___  sinh/p  1 

4:77  ds  L         cosh  £0  —  cos  (770  —  77)]  ' 

and  the  total  charge  is  zero.  When  the  total  charge  per  unit  length  is  1, 
and  the  total  charge  per  unit  length  of  the  line  is  —  1,  the  surface  density 
of  the  charge  on  the  cylinder  is 

1   drj  sinh  £0 

,  27T  ds  cosh  £0  —  cos  (779  —  77)  " 

This  is  what  C.  Neumann*  calls  the  induced  charge  density  or  the 
induced  loading;  it  represents,  of  course,  the  charge  on  one  side  of  the 
plate  £  =  0.  We  shall  write  this  expression  in  the  form 

<r(0,  77;  £)>??o)  ^^TT^fe  /S^°'7?;  ^°'7?0^ 
and  shall  use  a  corresponding  expression 

<*(fi>ih;  £o»?7o)  =  27T  ~dss  (£i>*?i;  £o>  ^o)* 

in  which 

<*  (t    «'£    »\  sinh  (|o  -  ^)  . 

s  (&,  ih,  6.  %)  =  cosh  ^--gy-.-ooa  &  -  ^)'      ......  (A) 

and  or  (fj,  77!;  ^0,  770)  is  the  density  of  the  induced  charge  for  the  elliptic 
cylinder  f  =  ^  . 

Let  us  now  consider  the  problem  in  which  a  function  V  is  required  to 
satisfy  the  condition  V  =  /  (77!)  on  the  cylinder  £  =  &,  while  V  is  a  regular 
potential  function  outside  the  cylinder  £  =  ^  but  not  necessarily  vanishing 
at  infinity.  Some  idea  of  the  nature  of  the  solution  may  be  obtained  by 
first  considering  the  two  cases 


Since 


^  cos  mr}9     V  ~  e~™<£-£i>  cos  W77, 
~  sin  77177,      V  —  e-"*(£-£i>  sin  77177. 


(&,  %;  f  ,  1?)  -  1  +  2  S  e-«««i>  cos  m  (77  -  77,), 

w-l 

*  Leipzig.  JBer.  Bd.  LXH,  S.  87  (1910). 

17 


258  Two-dimensional  Problems 

the  solution  is  given  in  these  cases  by  the  formula 

V  =  -S  (&,  %;  &  itf/fa)  Ah  ......  (B) 


and  we  may  write 

o-  (&,  *h;  £,  i?)  =  o-  (£1,  i?i)  5  (&,  ^;  f,  77),  ......  (C) 

where  o>(^1,  77^  is  the  natural  density  per  unit  length  when  the  total 
charge  per  unit  length  on  the  cylinder  f  =  ^  is  unity.  This  is  a  particular 
case  of  a  general  theorem  due  to  C.  Neumann*,  which  tells  us  that  the 
density  of  the  induced  charge  for  a  cylinder  whose  cross-section  is  a  closed 
curve  can  be  found  when  the  natural  density  on-  the  cylinder  and  the 
corresponding  potential  is  known.  The  expression  for  the  induced  charge 
is  then  of  the  form  (C),  where  £  and  77  are  conjugate  functions  such 
that  f  =  constant  are  the  equipotentials  and  77  =  constant,  the  lines  of 
force  associated  with  the  natural  charge.  The  undetermined  constant 
factor  occurring  in  the  expressions  for  functions  £  and  77  which  satisfy  the 
last  condition  should  be  chosen  so  that  77  increases  by  2?r  in  one  circuit 
round  the  cross-section  of  the  cylinder. 

The  formula  (A)  gives  a  potential  which  satisfies  the  conditions  of 
the  problem  for  a  wide  class  of  functions  and  for  this  class  of  functions 
we  have  the  interesting  relation 

27r  f  M  -  Urn  f  ^      Sinh  (^  ~~  &)/  foi)  *h         (f^f\ 
2w/(,)  -  hm  Jo   cogh  (f  _  ^  _  cog  (i-—j     (f  >  £). 

The  question  naturally  arises  whether  the  function  F  given  by  (B) 
is  the  only  function  which  fulfils  the  conditions  of  the  problem.  To  discuss 
this  question  we  shall  consider  the  case  when  the  ellipse  f  =  f  x  reduces 
to  the  line  £  =  0,  i.e.  the  line  S1S2. 

It  will  be  noticed  that  when  f  (rj)  =  1  the  formula  (B)  gives  7=1. 
Now  the  potential  <f>  which  is  the  real  part  of  the  expression 

(f>  +  i*ft=z  (z2  -  c2)~i  -  coth  £, 

satisfies  the  condition  that  </>  =  0  on  the  line  £  =  0  and  </>  =  1  at  infinity. 
Furthermore,  the  function  <f>lt  which  is  the  real  part  of 

<£i  -f  i^i  =  c  (z2  —  c2)"^  =  cosech  £, 
satisfies  the  conditions 

</»!  =  0  when  £  =  0,     fa  =  0  when  £  =  oo. 

Hence  a  more  general  potential  which  satisfies  the  same  conditions 

as  v  is  TT        j  i        -.-^  * 

F+  A<f>  +  B<f>l9 

*  Leipzig.  Ber.  Bd.  LXII,  S.  278  (1910). 


Munk^s  Theory  of  Thin  Aerofoils  259 

where  A  and  B  are  arbitrary  constants.  Now 

sinh  £  ^  1  4-  e-*+t*o 

cosh  £  —  cos  (T?  —  770)  ~~       1  —  e-£+<7>o 

__  o  sinh  $  -f  i  sin  T/O 
cosh  £  —  cos  170  * 
Hence,  if 

*    fsinh  £  -f  i  sin  770     cosh  £  —  A 


If*    f 
- 

2  TT  J  _„  [ 


-  ,  -  ,  —    --     -T— 

2  TT  J  _„  [  cosh  £  -  cos  770  smh  £ 

......  (D) 

where?/  and  F  are  constants,  the  potentials  u  and  v  are  conjugate  functions 
which  can  be  regarded  as  component  velocities  in  a  two-dimensional  flow 
of  an  incompressible  in  viscid  fluid.  These  component  velocities  satisfy  the 
conditions 

u  —  U,     v  =  F  at  infinity,     v  —  /  (77)  on  the  line  $x$2. 

This  result  is  of  some  interest  in  connection  with  Munk's  theory  of 
thin  aerofoils.  In  this  theory  an  element  ds  of  a  thin  aerofoil  in  a  steady 
stream  of  velocity  U  parallel  to  Ox  is  supposed  to  deflect  the  air  so  as  to 

give  it  a  small  component  velocity  v  =  u  -~  in  a  direction  parallel  to  the 

Ci  XQ 

axis  of  y.  Assuming  that  u  =  U  -f-  c,  where  e  is  a  small  quantity  of  the 
same  order  of  smallness  as  yQ  and  dy0/dx0  ,  we  neglect  e  ,  -  ,  as  it  is  of  order 

CiXQ 

e2,  and  write  v  =  U  -~-  .   This  is  now  taken  to  be  the  ^-component  of 

(LXq 

velocity  at  points  of  the  line  S1S2  and  the  corresponding  component 
velocities  (u,  v)  for  the  region  outside  the  aerofoil  may  be  supposed,  with 
a  sufficient  approximation,  to  be  given  by  an  expression  of  type  (D). 
In  this  expression,  however,  the  coefficient  A  is  given  the  value  1  so  that 
the  velocity  at  the  trailing  edge  will  not  be  infinite. 
Now  when  |  £  |  is  large  we  may  write 

(cosh  £  -  cos  Tjo)-1  =  sech  £  -f-  cos  TJQ  sech2  £  + 

sinh  £  =  cosh  £  —  |  sech  £  —  J  sech3  £  -f-  ...  , 
cosech  £  =  sech  £  -f  J  sech3  £  -f  ____ 
Hence,  when  V  =  0  the  flow  at  a  great  distance  from  the  origin  is 

°f  tyPe  V+iu=  iU  +  ft/*  +  &22  4    ...  , 

c   fir 
where  fa  =  2^  j  ^  (  1  -f  cos  770  -f  i  sin  rjQ)  f  fa)  drjQ  , 

c2  (n 
&  =  2-  J  _  (*  sin  770  cos  770  -  sin2  770)  /  (770)  <fy0  , 

17-3 


260  Two-dimensional  Problems 

and  by  Kutta's  theorem  the  lift,  drag  and  moment  per  unit  length  of  the 
aerofoil  are  given  by  the  expressions  of  §  4-  71 

L  +  iD  =  \p  [(v+  iu)2  dz^ 

M  =  \pR  f  (v  -f  w)2  zdz  =  - 
Therefore  L  -  -  PcU2  f*    (1  +  cos  T?O)  J?y°  cfy0, 

J   -  7T  ^#0 

sin  =0, 


These  are  the  expressions  obtained  by  Munk*  by  a  slightly  different 
form  of  analysis.  A  more  satisfactory  theory  of  thin  aerofoils  in  which  the 
thickness  is  taken  into  consideration,  has  been  given  by  Jeffreys  and  is 
sketched  in  §  4-73. 

Since  dxQ  =  —  c  sin  f]QdrjQ,     XQ  =  c  cos  T?O, 


we  may  write  L  -  -  2pU2  [    (c  +  XQ)  (c2  -  x^  -^  dxQ 

J  -c  uX0 

»c 

=      2pcU2\     (c  +  x0)-i(c-a;0Hy0^0, 

J  ~C 

M  =      2pU2  I      (c2  -  ^02)-*  xyodx0f. 

J  -c 

§  3-81.  Bipolar  co-ordinates.  Problems  relating  to  two  circles  which 
intersect  at  two  points  S1  and  $2  with  rectangular  co-ordinates  (c,  0), 
(—  c,  0)  revspectively  may  be  treated  with  the  aid  of  the  conformal 

transformation  .  .  .,  .  .  v 

z  =  ic  cot  |^,  ......  (A) 

where  z  ^  x  +  iy,   £  =  £  4-  ^  and  (^,  t/),   (f  ,  7^)  are  the  rectangular  co- 
ordinates of  two  corresponding  points  P  and  TT.    We  shall  say  that  the 
point  P  is  in  the  z-plane  and  the  point  TT  in  the  £-plane.  The  transformation 
may  be  said  to  map  one  plane  on  the  other. 
It  is  easily  seen  that 


z  -  c 


where 


=  (x  _  c)2  -f  s/2  -  I  z  -  c  j  2  -  2cMe~\ 
=  (a:  +  c)2  -f  7/2  =  |  z  +  c  |2 


=  -... 

cosh  77  —  cos  f 

*  National  Advisory  Committee  for  Aeronautics,  Report,  p.  191  (1924);  see  also  J.  S.  Ames, 
Report,  p.  213  (1925)  and  C.  A.  Shook,  Amer.  Journ.  Math.  vol.  XLvm,  p.  183  (1926). 


Bipolar  Co-ordinates  261 

The  curves  £  =  constant  are  clearly  circles  through  the  points  8l  and 
S2,  while  the  curves  77  =  constant  p 

are  circles  having  these  points  as 
inverse  points.  The  two  sets  of 
curves  form  in  fact  two  orthogonal 
systems  of  circles,  as  is  to  be  ex- 
pected since  the  transformation  (A) 
is  conformal,  and  the  corresponding 
sets  of  lines  are  perpendicular.  S2 

The  expressions  for  x  and  y  in  Fis-  1()- 

terms  of  f  and  rj  are  x  =  M  sinh  77,  y  =  M  sin  f  .   At  a  point  P0  of  the 
line  ^$2  we  have  £  =  TT,  therefore 

#0  -  c  tanh 


and  the  natural  loading  for  this  line  is 

o-0=  (I/we)  cosh  (iyo/2  )• 

The  loading  induced  by  a  charge  —  1  at  the  point  P  (f  ,  r?)  is,  on  the 
other  hand, 


_ 

°  7T0  [cosh  (77  —  7?0)  -f  cos  £] 

cosh  fo/2)  +  cosh  [fo/2)  -  770]         f 

—    CTn  |       ,  >  ,.  UUO    rt  • 

cosh  (Y]  —  T?O)  -f  cos  f  2 

EXAMPLE 

A  potential  function  v  is  regular  in  the  semicircle  y  >  0,  x*  -f  y2  <  a2  and  satisfies  the 
boundary  conditions  v  =  A  when  t/  =-  0,  ^   -=  .B  when  x2  +  y2  —  a2,  prove  that 


/(JO 

v  =  A-A' 
Jo 


(£      TT)        dm 

m  ,  OWTT 

cosh- 

where  x  -f  t/y  =  ia  cot  j  (£  -f  *„),  .4'  =  a#. 

§  3-82.   Effect  oj  a  mound  or  ditch  on  the  electric  potential.   Let  us  now 
consider  the  complex  potential 

2c         £ 

where  K  is  a  real  constant  at  our  disposal.  The  potential  <f>  is  zero  when 
£  =  0,  for  in  this  case  ^  becomes  cot  ,  and  is  a  purely  imaginary 

K.  /C 

quantity.    It  is  also  zero  when  £  =  |/CTT,  for  then  #  = tan  ^ ,  and  is 

again  a  purely  imaginary  quantity.  The  potential  <£  is  thus  zero  on  a 
continuous  line  made  up  ,of  the  portion  of  the  line  y  =  0  outside  the 
segment  S1S2  and  of  the  circular  arc  through  S1S2  at  points  of  which 


262  Two-dimensional  Problems 

S1S2  subtends  the  angle  \KTT.  Thus  the  complex  potential  x  provides  us 
with  the  solution  of  an  electrical  problem  relating  to  a  conductor  in  the 
form  of  an  infinite  plane  sheet  with  a  circular  mound  or  ditch  running 
across  it. 

<,.  d(f>      (cosh  r)  —  cos  f  )2  dc/> 

dy      cosh  77  cos  £  —  1    9f  ' 

we  find  that  on  the  axis  of  y,  where  77  =  0, 


Al  fy  2C  of 

Also  -£  =  -    -„  cosec2  -, 

d£  /c2  AC 


consequently  the  potential  gradient  on  the  axis  x  =  0  is 

2  £ 

2  cosec2  -  (1  —  cos  |). 

At  the  vertex  where  £  =  |KTT  it  is 

2  (1  —  cos  I/CTT). 

On  the  plane  y  =  0,  we  have  f  -  0,  and  the  gradient  is 
„  cosech2  -.(cosh  -n  —  I). 

K*  K 

As  ??  -vQ,  x->co  and  the  gradient  tends  to  the  value  1  which  will  be 
regarded  as  the  normal  value. 

As  77  ->  ±  oo,  x  ->  ±  c  and  the  gradient  tends  to  become  zero  or  infinite 
according  as  K  ^  2. 


Fig.  17.  Fig.  18. 

When  K  —  1  we  have  a  semicircular  mound. 

The  gradient  on  the  line  x  =  0  is  everywhere  greater  than  the  normal 
value,  at  the  vertex  it  is  2,  and  at  a  point  at  distance  2c  above  the  vertex 
it  is  10/9. 

When  AC  —  3  we  have  a  semicircular  ditch. 

The  gradient  on  the  line  x  =  0  is  everywhere  less  than  the  normal 
value,  at  the  bottom  of  the  ditch  it  is  2/9  and  at  a  point  (0,  c),  at  distance 
2c  above  the  bottom,  it  is  8/9.  By  making  K  ->  0  we  obtain  values  of  the 


.     CL7T 

v  cosec2 


Electrical  Effects  of  Peaks  and  Pits  263 

gradient  for  the  case  of  a  cylinder  standing  on  an  infinite  plane.  We  must 
naturally  make  c  -*•  0  at  the  same  time,  in  order  to  obtain  a  cylinder  of 
finite  radius  a.  The  appropriate  complex  potential  is 

X  =  <f>  +  i$  =  an  cot  ---  .   .  ......  (C) 

On  the  line  x  =  0  the  potential  gradient  is 

CL7T\ 

, 

yr 

and  tends  to  the  normal  value  as  y  ->  ±  oo. 

When  T/  ==  2a  the  gradient  is       ,  which  is  nearly  2-5.  At  a  distance  2a 

2 

above  the  summit,  y  =  4a  and  the  gradient  is  ^  =  1-2337.    On  the  axis 

8 

of  x  the  gradient  is  0   0 

a7T  i**fa7r\ 

—  0    cosech2 

x2  \  x  J 

As  x  ->  0  the  gradient  diminishes  rapidly  to  zero,  consequently  the 
surface  density  of  electricity  is  very  small  in  the  neighbourhood  of  the 
point  of  contact. 

EXAMPLE 

Fluid  of  constant  density  moves  above  the  infinite  plane  y  =  0  with  uniform  velocity  U. 
A  cylinder  of  radius  a  is  placed  in  contact  with  the  plane  with  its  generators  perpendicular 
to  the  flow.  Prove  that  the  stream  function  is  derived  from  a  complex  potential  of  type  (C) 
multiplied  by  U  and  calculate  the  upward  thrust  on  the  cylinder. 

[H.  Jeffreys,  Proc.  Camb.  Phil.  Soc.  vol.  xxv,  p.  272  (1929).] 

§  3*83.  The  effect  of  a  vertical  wall  on  the  electric  potential.  Let  h  be  the 
height  of  the  wall,  x  =  </>  +  ty  the  complex  potential.  If  a  is  a  constant, 

the  substitution  .,  i  m 

az  =  ih  (a2  +  x)  ......  (**) 

makes  the  point  z  =  ih  correspond  to  x  =  0>  the  points  on  the  axis  of  x 
correspond  to  the  points  on  </>  =  0  for  which  02  >  a2,  while  the  points  on 
the  axis  of  y  for  which  y  <  h  correspond  to  the  points  on  <f)  =  0  for 
which  02  <  a2.  Hence,  if  </>  be  regarded  as  the  electric  potential,  a  con- 
ducting surface  consisting  of  the  plane  y  =  0  and  the  conducting  wall 
(x  =  0,  y  <  h)  will  be  at  zero  potential*. 

If  (r,  0),  (/,  6')  are  the  bipolar  co-ordinates  of  a  point  P  relative  to  $, 
the  top  of  the  wall,  and  to  AS',  the  image  of  this  point  in  the  plane  y  =  0, 
we  have 

h*x2  =  -  a2  (z2  +  h*)=  -  a2rr'ei(e+e'>. 
Therefore  h<f>  =  a  (rrr$  sin  J  (6  4-  0')> 

Jufi  -  -  a  (rr')i  cos  £  (0  +  (9'). 

Therefore     2Afy2  -  a2  {[(x2  -  i/2  +  A2)2  +  4x2y2]*  -  (x2  -  y2  H-  A2)}. 
*  G.  H.  Lees,  Proc.  Roy.  Soc.  London,  A,  vol.  xci,  p.  440  (1915), 


264  Two-dimensional  Problems 

The  equipotentials  have  been  drawn  by  Lees  from  the  equation 

y2  (1  +  a2x2/h2cf>2)  =  h2  +  x2  -f  h2<f>2/a*. 

To    determine    the    surface    density    of   electricity   we    differentiate 
equation  (D),  then 


When  a:  —  0,  z  =  it/,  /^  =  —  ia  (A2  —  y2)^,  and  so 


The*  surface  density  is  thus  zero  at  the  base  and  infinite  at  the  top  of 
the  wall. 

When  y  =  0,  z  =  x,  />#  =  —  ia  (h2  -f  x2)^,  and  so 


As  a;  ->  oo  this  tends  to  the  value  a/h  which  may  be  regarded  as  the 
normal  value  of  the  gradient.  At  a  distance  from  the  foot  equal  to  h  the 
vertical  gradient  is  0-707  times  the  normal  gradient. 

The  curve  along  which  the  electric  field  strength  has  the  constant  value 
F  is  -given  by 

a2  (x2  +  y2)  -  h2F2  [(x2  -  y2  +  h2)2  + 

that  i8' 


where  (7?,  0)  are  the  polar  co-ordinates  of  P  with  respect  to  the  origin. 
The  curves  F  =  constant  may  be  obtained  by  inversion  from  the  family 
of  Cassinian  ovals  with  8  and  $'  as  poles,  they  are  the  equipotential  curves 
for  two  unit  line  charges  at  S  and  $',  and  a  line  charge  of  strength  —  2  at 
the  origin  0.  The  rectangular  hyperbola 

7/2  -  X2  -   |/*2 

is  a  particular  curve  of  the  family.  This  hyperbola  meets  the  axis  of  y  at 
a  point  where  the  horizontal  gradient  is  equal  to  the  normal  gradient. 
The  force  is  equal  in  magnitude  to  the  normal  gradient  at  all  points  of 
this  hyperbola.  At  points  above  the  hyperbola  the  force  is  greater  than 
a/h,  at  points  below  the  hyperbola  it  is  less  than  a/h. 

The  curves  along  which  the  force  has  a  fixed  direction  are  the  lemnis- 
cates  defined  by  the  equation 

0  +  0'  —  20  =  constant. 

Each  lemniscate  passes  through,  S  and  S'  and  has  a  double  point  at  0. 
It  should  be  noticed  that  the  transformation 

az  -  ih  (a2  -  & 
enables  us  to  map  the  upper  half  of  the  £-plane  on  the  region  of  the  upper 


Effect  of  a  Vertical  Wall  265 

z-plane  bounded  by  the  line  y  =  0  and  the  vertical  wall  x  =  0,  y  <  h.  This 
transformation  makes  the  points  at  infinity  in  the  two  planes  corre- 
sponding points.  It  may  be  observed  also  that  if  a2  —  £2  —  r2e2ie,  the 
angle  20  ranges  from  —  TT  to  TT.  Hence,  since  az  =  hr  sin  6  -f  ihr  cos  0, 
hr  cos  0  is  never  negative  and  so  it  is  the  upper  portion  of  the  cut  z-plane 
Wjhich  corresponds  to  the  upper  portion  of  the  £-plane.  If  we  invert  the 
z-plane  from  a  point  on  the  negative  portion  of  the  axis  of  y  we  obtain 
a  region  inside  a  circle  which  is  cut  along  a  radius  from  a  point  on  the 
circumference  to  a  point  not  on  the  circumference.  The  upper  half  of  the 
£-plane  maps  into  the  interior  of  this  region. 

If,  on  the  other  hand,  we  invert  from  the  origin  of  the  z-plane,  the 
cut  upper  half  plane  inverts  into  a  half  plane  with  a  cut  along  the  y-axis 
from  infinity  to  a  point  some  distance  above  the  origin.  The  point  at 
infinity  in  the  £-plane  now  maps  into  the  origin  in  the  z-plane. 


CHAPTER  IV 

CONFORMAL  REPRESENTATION 

§  4§  11.  Many  potential  problems  in  two  dimensions  may  be  solved 
with  the  aid  of  a  transformation  of  co-ordinates  which  leaves  V27  =  0 
unaltered  in  form.  It  is  easily  seen  that  the  transformation 

£  =  /  (*>  y)>   i?  =  9  (x>  y) 

furnished  by  the  equation 

l=g  +  ii,  =  F(x  +  iy)  =  F  (z) 

possesses  this  property  when  the  function  F  is  analytic,  because  a  function 
of  £  which  is  analytic  in  some  region  F  of  the  £-plane  is  also  analytic  in 
the  corresponding  region  G  of  the  z-plane  when  regarded  as  a  function  of  z. 
In  using  a  transformation  of  this  kind  it  is  necessary,  however,  to  be 
cautious  because  singularities  of  a  potential  function  may  be  introduced 
by  the  transformation,  and  the  transformation  may  not  always  be  one-to- 
one,  i.e.  a  point  P  in  the  £-plane  may  not  always  correspond  to  a  single 
point  Q  in  the  z-plane  and  vice  versa. 

Let  F  be  a  function  of  f  and  77  which  is  continuous  (D,  1),  then 

37^373£     aFcfy       37^3731     dVdrj 
dx       3£  dx      drj  3xJ        dy       dg  dy      877  dy' 

These  equations  show  that  if  the  derivatives  of.  £  and  77  are  not  all 
finite  at  a  point  (x,  y)  in  the  z-plane,  the  derivatives  ~— ,  ~  may  be  infinite 

37  dV 

even  though  -^  and    _      are  finite.    A  possible  exception  occurs  when 

3F         37 

yr  and  -x     both  vanish,  i.e.  at  a  point  of  equilibrium  or  stagnation. 

At  any  point  (#0,  yQ)  in  the  neighbourhood  of  which  the  function  F  (z) 
can  be  expanded  in  a  Taylor  series  which  converges  for  |  z  —  z0  |  <  c, 
we  have 

F  (z)  =  1  an  (z  -  z0)», 

n-O 

where  z  =  x  +  iy,     z0  =  XQ  +  iy0 , 

and  if  £  =  f  +  i,  =  J  (z),     £0  =  £0  +  irjQ  =  F  (z0), 

we  may  write 

d£  -  £  -  £0  =  F  (z)  -  ^  (z0)  =  rfz  [f  (z)  +  e], 
where  rfz  =  z  —  z0  and  e  ->  0  as  dz  ->  0. 

Hence  d£  =  dz.^1'  (x)  approximately. 


Properties  of  the  Mapping  Function  267 

This  relation  shows  that  the  (x,  y)  plane  is  mapped  conformally  on  the 
(f ,  rj)  plane  for  all  points  at  which  |  F'  (z)  \  is  neither  zero  nor  infinite. 
We  have  in  fact  the  approximate  relations 

da  =  ds  |  F'  (z)  | , 
<f>=  6+  a, 
where  dz  =  ds.eie,     dt,  =  dv.e^, 

F'  (z)  =  |  F'  (z)  |  4*. 

These  relations  show  that  the  ratio  of  the  lengths  of  two  corresponding 
linear  elements  is  independent  of  the  direction  of  either  and  that  the  angle 
between  two  linear  elements  dz,  8z  at  the  point  (x,  y)  is  equal  to  the  angle 
between  the  two  corresponding  linear  elements  at  (£,  77).  The  first  angle  is, 
in  fact,  6  —  6' ',  while  the  second  angle  is 

<!>-<l>'=(0  +  a)-(8'  +  a)  =  0-  6'. 

These  theorems  break  down  if  some  of  the  first  coefficients  in  the 
expansion 

£-  ^^^an(z~z.Y  (A) 

n-l 

are  zero.  If,  for  instance,  a^  =  a2  =  ...  am^  =  0,  we  have  for  small  values 

Of    |    Z    -    ZQ    | 

£  -  £o  =  am  (*  ~  Zo)m> 
and  the  relation  between  the  angles  is 

,     <f>  =  m0  +  am,  where  am=  \am\  e"m. 
This  gives 

<f>  -  </>'  -  m  (0  -  6'). 

More  generally  if  there  is  an  expansion  of  type  (A)  in  which  the 
lowest  index  m  is  not  an  integer  a  similar  relation  holds. 

§  4- 12.  The  way  in  which  conformal  representation  may  be  used  to 
solve  electrical  and  hydrodynamical  problems  is  best  illustrated  by  means 
of  examples.  One  point  to  be  noticed  is  that  frequently  the  transformation 
does  not  alter  the  essential  physical  character  of  the  problem  because  an 
electric  charge  concentrated  at  a  point  (line  charge)  corresponds  to  an 
equal  electric  charge  concentrated  at  the  corresponding  point,  a  point 
source  in  a  two-dimensional  hydrodynamical  problem  corresponds  to  a 
point  source  and  so  on. 

These  results  follow  at  once  from  the  fact  that  if  <f>  +  i$  is  the  complex 
potential  we  may  write 

<t>  +  i*f>=f(x+  iy)  -  g  (f  +  iy), 

and  if  <f>  is  the  electric  potential,  the  integral  \difj  taken  round  a  closed  curve 
is  ±  47T  times  the  total  charge  within  the  curve.    Now  the  interior  of  a 


268  Confonnal  Representation 

closed  curve  is  generally  mapped  into  the  interior  of  a  corresponding  closed 
curve  and  a  simple  circuit  generally  corresponds  to  a  simple  circuit, 
moreover  0  is  the  same  in  both  cases  and  so  the  theorem  is  easily  proved. 
It  should  be  noted  that  a  simple  circuit  may  fail  to  correspond  to  a  simple 
circuit  when  the  closed  curve  contains  a  point  at  which  the  conformal 
character  of  the  transformation  breaks  down.  Another  apparent  exception 
arises  when  a  point  (x,  y)  corresponds  to  points  at  infinity  in  the  (£,  rj) 
plane,  but  there  is  no  great  difficulty  if  these  points  at  infinity  are  imagined 
to  possess  a  certain  unity.  In  fact  mathematicians  are  accustomed  to 
speak  of  the  point  at  infinity  when  discussing  problems  of  conformal 
representation.  This  convention  is  at  once  suggested  by  the  results  obtained 
by  inversion  and  is  found  to  be  very  useful.  There  is  no  ambiguity  then 
in  talking  of  a  point  charge  or  source  at  infinity. 

We  have  s§en  that  certain  angles  are  unaltered  by  a  conformal  trans- 
formation and  can  consequently  be  regarded  as  invariants  of  the  trans- 
formation. Certain  other  quantities  are  easily  seen  to  be  invariants. 
Writing 


where  d  (x,  ?/),  d  (£,  77)  are  elements  of  area  in  the  two  planes,  we  have 


J  (       _L     -          r  \  -    ^    ^  _i_ 

W  '"dr"'1      °" 


J 


Sx 


The  quantities  </>  and  0  are  usually  taken  to  be  invariants  in  a  con- 
formal  transformation  and  the  foregoing  relations  indicate  that 


*   ffl"V9<A\2     /3^\21  7  ^ 

.  and    \\\(J}   +U       \dxdy 

cx2        y2/  }}[\dxj       \dyj  J        * 

are  invariants.  In  the  theory  of  electricity  the  first  integral  is  proportional 
to  the  total  charge  associated  with  the  area  over  which  the  integration 
takes  place.  In  hydrodynamics  the  second  integral  represents  the  total 
vorticity  associated  with  the  area  and  the  third  integral  is  proportional 
to  the  kinetic  energy  when  the  density  of  the  fluid  is  constant.  The 

invariant  character  of  the  integrals  (udx  -f  vdy)  and  (udy  —  vdx)  is  easily 
recognised  because  these  represent  \d(f>  and  \difs  respectively. 


Riemann  Surfaces  and  Winding  Points  269 

§  4*21.  The  transformation  w  =  zn.  When  n  is  a  positive  integer  the 
transformation  w  =  zn  does  not  give  a  (1,  1)  correspondence  between  the 
w-plane  and  the  z-plane  but  it  is  convenient  to  consider  an  ti-sheeted 
surface  instead  of  a  single  plane  as  the  domain  of  w.  For  a  given  value  of 
w  the  equation  zn  —  w  has  n  roots.  If  one  of  these  is  Zl  the  others  are 
respectively  Z2  =  Z^,  Z3  =  Z^2,  ...  Zn  =  21o>w-1,  where  oj  =  e2rrt/n. 

If  2  =  reie  we  may  adopt  the  convention  that  for 

Zly     0<  710  <  2,T, 
Z2,     2n<  n9<  477, 


Zn,        (2W  —    2)   77  <    710  <    2/177. 

Defining  the  sheet  (m)  to  be  that  for  which  Wm  =  Zmn  we  can  say  that 
W±  is  in  the  first  sheet,  PF2  in  the  second  sheet,  and  so  on.  The  n  sheets 
together  form  a  "Riemann  surface"  arid  we  can  say  that  there  is  a  (1,  1) 
correspondence  between  the  z-plane  and  the  Riemann  surface  composed 
of  the  sheets  (1),  (2),  ...  (n).  If  w  --=  Re1®  we  have  0  =  n9,  and  so  when 
w  =  W^n»  we  have  (2m  —  2)  77  <  0  <  2w7r. 

The  z-plane  is  .divided  into  n  parts  by  the  lines  joining  the  origin  to 
the  corners  of  a  regular  polygon,  one  of  whose  corners  is  on  the  axis  of  x. 
These  n  portions  of  the  z-plarie  are  in  a  (1,  1)  correspondence  with  the 
n  sheets  of  the  Riemann  surface.  The  n  lines  just  mentioned  each  belong 
to  two  portions  and  so  correspond  to  lines  common  to  two  sheets.  It  is  by 
crossing  these  lines  that  a  point  passes  from  one  sheet  to  another  as  the 
angle  0  steadily  increases.  The  point  O  in  the  w-plane  is  a  winding  point 
of  the  Riemann  surface,  its  order  is  defined  as  the  number  n  —  1. 

A  circle  |  w  —  W  \  =  an  corresponds  to  a  curve  |  zn  —  Zn  \  =  an,  which 
belongs  to  the  class  of  lemniscates* 

^2  ...  rn  =  an, 

where  rx ,  r2 , . . .  rn  are  the  distances  of  the  point  z  from  the  points  Zx ,  Z2 , . . .  Zn 
which  correspond  to  W.  In  the  present  case  the  poles  of  the  lemniscate 
are  at  the  corners  of  a  regular  polygon  and  the  equation  of  the  lemniscate 
can  be  expressed  in  the  form 

r2n  _  2rnSn  cos  n  (0  -  0)  +  R2n  =  a2n     (reie  =  z,  Rei&  =  Z). 

When  n  =  2  a  circle  in  the  z-plane  corresponds  to  a  lima^on.  To  see 
this  we  write  w  =  u  +  iv,  z  =  x  +  iy,  then 

u  =  x2  —  y2y     v  =  2xy. 

*  This  is  the  name  used  by  D.  Hilbert,  Gutt.  Nachr.  S.  63  (1897).  The  name  cassinoid  is  used 
by  C.  J.  de  la  Vallee  Poussin,  Mathesis  (3),  t.  n,  p.  289  (1902),  Appendix.  The  geometrical  pro- 
perties and  types  of  curves  of  this  kind  are  discussed  by  H.  Hilton,  Mess,  of  Math.  vol.  XLVIII, 
p.  184  (1919),  reference  being  made  to  the  earlier  work  of  Serret,  La  Goupilliere  and  Darboux. 


270  Conformal  Representation 

Hence,  if  .          ,0        0        0 

(x  +  a}*  +  y2  =  c2, 

we  have  [u2  4-  v2  -  2a2u  +  (a2  -  c2)2]2  -  4c4  (^2  +  v2), 

or,  if        /7  -  u  +  c2  -  a2,     F  =  v,     U  ^  RcosQ,     V  ^  R  sin  0, 
([72  +  72  „  2c2Z7)2  -  4a2c2  (?72  +  F2), 
R  =  2ac  +  2c2cos0. 

EXAMPLES 

1.  The  curve  r2n  —  2rncn  cos  n0  -f-  cndn  —  0  has  w  ovals  each  of  which  is  its  own  inverse 
with  respect  to  a  circle  centre  O  and  radius  \/(cd).  The  ordinary  foci  Bl9B2,...Bn  invert  into 
the  singular  foci  A19  A2,  ...  An,  the  polar  co-ordinates  of  Bs  being  given  by  r  ^  d,  nd  —  ZSTT. 

2.  Line  charges  of  strength  4-  1  are  placed  at  the  corners  of  a  regular  polygon  of  n 
corners  and  centre  0,  while  line  charges  of  strength  —  1  are  placed  at  the  corners  of  another 
regular  polygon  of  n  corners  and  centre  O.  Prove  that  the  equipotentials  are  w-poled  lemnis- 
cates.  [Darboux  and  Hilton.] 

3.  Prove  also  that  the  lines  of  force  are  n-poled  lemniscates  passing  through  the  vertices 
of  the  regular  polygons. 

§  4-22.   The  bilinear  transformation.   The  transformation 

r  ^  az±b  (A} 

^~    az+$>  (A) 

in  which  a,  6,  a,  ]3  are  complex  constants,  is  of  special  interest  because  it 
is  the  only  type  of  transformation  which  transforms  the  whole  of  the 
z-plane  in  a  one-to-one  manner  into  the  whole  of  the  £-plane  and  gives  a 
conformal  mapping  of  the  neighbourhood  of  each  point. 

If  a  i-  0  there  are  generally  two  points  in  the  z-plane  for  which  £  =  z. 
These  are  given  by  the  quadratic  equation 

az2  +  (j8  -  a)  z  -  b  -  0. 

Let  us  choose  our  origin  in  the  z-plane  so  that  it  is  midway  between 
these  points,  then  /?  =  a  and  if  we  write  b  =  ac2  the  self-corresponding 
points  are  given  by  z  =  ±  c.  The  transformation  may  now  be  written  in 

theform  £+_c._a  +  ca«  +  c 

£  —  c     a  —  caz  ~  c ' 

From  this  relation  a  geometrical  construction  for  the  transformation 
is  easily  derived.  Writing  a  +  ca  ^  ;„ 

£  +  c  =  RI&*I,    z  +  c  -  r^'i, 
£  —  c  =  R2el®2,    z  —  c  =  r2eiezr 

R         r 

we  have  the  relations  n1  =  p  -1 , 

^2        *> 

0!  -  0,  =  CO  +    (^  -  *,). 


The  Bilinear  Transformation  271 

If  Sl  and  S2  are  the  self  -corresponding  points  these  relations  tell  us  that 
a  circle  through  Sl  and  S2  generally  corresponds  to  a  circle  through  Sl 
and  $2,  but  in  an,  exceptional  case  it  may  correspond  to  a  straight  line, 
namely  the  line  $!&,. 

Again,  a  circle  which  has  Sl  and  S2  as  inverse  points  corresponds  to  a 
circle  which  has  $x  and  S2  as  inverse  points. 

By  a  suitable  displacement  of  the  z  and  £-planes  we  can  make  any 
given  pair  of  points  the  self-corresponding  points  provided  the  self-corre- 
sponding points  &re  distinct,  for  if  the  displacements  are  specified  by  the 
complex  quantities  u  and  v  respectively,  the  transformation  may  be 
written  in  the  form 


and  we  can  choose  u  and  v  so  that  the  equation  £  =  z  has  assigned  roots 
zl  and  z2. 

We  may  conclude  from  this  that  the  transformation  maps  any  circle 
into  either  a  straight  line  or  a  circle;  a  result  which  may  be  proved  in 
many  ways.  One  proof  depends  upon  the  theorem  that  in  a  bilinear 
transformation  of  type  (A)  the  cross-ratio  of  four  values  of  z  is  equal  to 
the  cross-ratio  of  the  four  values  of  £;  i.e. 

(z-  z,)  (*8  -_z3)   (r^j^Hk  -  k) 

(*  -  z2)  (*3  -  *i)    (£  -  £2)  (&  -  £iV 

Now  the  cross-ratio  is  real  when  the  four  points  lie  on  either  a  straight 
line  or  a  circle,  hence  four  points  on  a  circle  must  map  into  four  points 
which  are  either  collinear  or  concyclic.  If  in  the  transformation  (A)  we 
choose  u  so  that  au  +  ft  =  0,  and  v  so  that  av  =  a,  the  transformation 

takes  the  form  y        1  9  ,._ 

£z=&2,  ......  (B) 

where  a2k2  —  ab  —  aft. 

By  a  suitable  rotation  of  the  axes  of  reference  we  can  reduce  the 
transformation  to  the  case  in  which  k  is  real,  and  this  is  the  case  which 
will  now  be  discussed. 

The  transformation  evidently  consists  of  an  inversion  in  a  circle  of 
radius  k  with  centre  at  the  origin  followed  by  a  reflection  in  the  axis  of  x. 
The  points  z  =  ±  k  are  self-corresponding  points  and  if  these  are  denoted 
by  S1  and  S2  it  is  easily  seen  that  two  corresponding  points  P  and  Q  lie 
on  a  circle  through  S^  and  S2.  The  figure  has  a  number  of  interesting 
properties  which  will  be  enumerated. 

1.  Since  z  (£  -f  k)  =  k  (z  -f  k)  the  angles  /SXPO,  QS^  are  equal,  and 
so  the  angles  S1PO,  S2PQ  are  also  equal. 

2.  The  triangles  S1PO,  QPS%  are  similar,  and  so 

PSi.PSt~PO.PQ. 


272  Conformal  Representation 

3.  If  C  is  the  middle  point  of  PQ  we  have  CS^CS^  =  CP2,  also  PQ 
bisects  the  angle  S^CS2. 

The  four  points  Sl9  P,  S2,  Q  on  the  circle  form  a  harmonic  set.  This 
follows  from  the  relation 

1     4-   _      =-r 

z_ k^z  + k     z- V 

which  is  easily  derived  from  (B). 

The  angle  PS^C  is  equal  to  the  angle  8^0.  The  lines  ^P,  P0,  (7^ 
thus  form  an  isosceles  triangle. 

In  the  case  when  the  self-corresponding  points  coincide  we  have 

a  -  fl  =  2ac,     b  =  -  ac2, 

where  c  is  the  self-corresponding  point.  The  transformation  may  now  be 
written  in  the  form 

~  =  Q  -"-  •--+  —  ,    £  +  ac  *  0. 
£—  c     ft  +  ac      z  ~  c      r 

It  may  be  built  up  from  displacements  and  transformations  of  the 
type  just  considered  and  so  needs  no  further  discussion.  The  only  other 
interesting  special  case  is  that  in  which  the  transformation  then  consists 
of  a  displacement  followed  by  a  magnification  and  rotation. 

§  4-23.  Poissorts  formula  and  the  mean  value  theorem.  Bocher  has 
shown  by  inversion  that  Poisson's  formula  may  be  derived  from  Gauss's 
theorem  relating  to  the  mean  value  of  a  potential  function  round  a  circle. 

Let  C  and  C'  be  inverse  points  with  respect  to  the  circle  F  of  radius  a, 
and  let  CO'  =  c.  Inverting  with  respect  to  a  circle  whose  centre  is  C'  and 
radius  c,  the  point  8  on  the  circle  transforms  into  a  point  8'.  We  shall 
suppose  that  C'  is  outside  the  circle  F,  then  S  is  inside  the  circle  F.  Let 
dsy  els'  be  corresponding  arcs  at  8  and  8'  respectively  and  let  the  polar 
co-ordinates  of  C  and  C1  be  (r,  6),  (/,  0)  respectively,  where  rr'  =  a2.  The 
circle  F  inverts  into  a  circle  with  centre  C  and  radius  given  by  the  formula 
aa'  -  or,  for  rR  „. 

PS'  ~  r  -  r  ° 

C£    -CC'S          CM 

a  —  r      cr        , 

=  c  -,—  -  =—  =  a  . 
r  —  a     a 

Also         r*C'S*  -  a*.CS2  -  a2  [r2  -fa2-  2ar  cos  (a  -  0)], 

where  (a,  a)  are  the  polar  co-ordinates  of  8. 
Writing  ds'  ~  a'dv  ,  we  have 

,  ,  _     c2ds         ca2da  __  rcda 

=  a'T  C"&»  ^  r ".  C'S*  =  a2  -  2ar  cos~(a  - "^"Tr5 ' 
and  re  =  a2  —  r2,  consequently  the  formula  of  Poisson  becomes 


Circle  and  Half  Plane  273 

This  formula  states  that  the  mean  value  of  a  potential  function  round 
the  circumference  of  a  circle  is  equal  to  the  value  of  the  function  at  the 
centre  of  the  circle.  Hence  Poisson's  formula  may  be  derived  from  this 
mean  value  theorem  and  is  true  under  the  same  conditions  as  the  mean 
value  theorem. 

§  4-24.  The  conformal  representation  of  a  circle  on  a  Jialf  plane*.  If  two 
plane  areas  A  and  A±  can  be  mapped  on  a  third  area  AQ  they  can  be 
mapped  on  one  another,  eonseqiiently  the  problem  of  mapping  A  on  At 
reduces  to  that  of  mapping  A  and  Al  on  some  standard  area  A0. 

This  standard  area  AQ  is  generally  taken  to  be  either  a  circle  of  unit 
radius  or  a  half  plane.  The  transition  from  the  circle  x2  -f  y2  <  1  in  the 
z-plane  to  the  half  plane  v  >  0  in  the  ?#-plane  is  made  by  means  of  the 
substitutions 

z  =  x  -f  iy,     w  =  u  -j-  iv,     z  (i  -j-  w)  =  i  —  w, 

Dx  -  1  -  u2  -  v2,     l)y  -  2i/,  (I) 

where  D  =  u2  {-  (1  -f-  v)2, 

4/D  -  (1  4-  x)2  -f  y2. 

When  v  =  0,  the  substitution  u  =  tan  6  gives 
x  =  cos  20,     y  =  sin  29. 

As  26  varies  from  —  TT  to  TT,  the  variable  u  varies  from  —  oo  to  oo  and 
so  the  real  axis  in  the  w-plane  is  mapped  in  a  uniform  manner  on  the  unit 
circle  x2  -f-  y2  =  1  in  the  z-plane. 

Since,  moreover, 

D  (1  -  x2  -  y2)  =  4?;,     Dy  =  2u9 

we  have  v  >  0  when  x2  -f  y2  <  1,  consequently  the  interior  of  the  circle  is 
mapped  on  the  upper  half  of  the  w-plane. 

When  u  and  v  are  both  infinite  or  when  either  of  them  is  infinite,  we 
have  x  =  —  1,  y  =  0;  hence  the  point  at  infinity  in  the  w-plane  corre- 
sponds to  a  single  point  in  the  z-plane  and  this  point  is  on  the  unit  circle. 

The  transformation  (I)  may  be  applied  to  the  whole  of  the  z-plane; 
it  maps  the  region  outside  the  circle  x2  +  y2  =  1  on  the  lower  half  (v  <  0) 
of  the  w-plane.  A  line  y  =  mx  drawn  through  the  centre  of  the  circle 
corresponds  to  a  circle  m  (1  —  u2  —  v2)  =  2u  whjch  passes  through  the 
point  (0,  1)  which  corresponds  to  the  centre  of  the  circle,  and  through  the 
point  (0,  —  1)  which  corresponds  to  the  point  at  infinity  in  the  z-plane. 
This  circle  cuts  the  line  v  =  0  orthogonally. 

Two  points  which  are  inverse  points  with  respect  to  the  circle  x2  -j-  y2  =  1 
map  into  points  which  are  images  of  each  other  in  the  line  v  =  0. 

*  This  presentation  in  §§  4-24,  4-61  and  4-62  follows  closely  that  given  in  Forsyth's  Theory  of 
Functions  and  the  one  given  in  Darboux's  Thdorie  generate  des  surfaces,  1. 1,  pp.  170-180. 
B  18 


274  Conformal  Representation 

The  upper  half  of  the  w-plane  may  be  mapped  on  itself  in  an  infinite 
number  of  ways.  To  see  this,  let  us  consider  the  transformation 

Y      aw  +  b  dt,  —  b  /T1A 

£  = -7-J9       W  =  — =,  (II) 

cw  -f  d  a  —  c£ 

in  which  a,  6,  c  and  d  are  real  constants  and  £  =  £  -f  irj. 

When  w  is  real  £  is  also  real  and  vice  versa,  hence  the  real  axes  corre- 
spond. Furthermore, 

7?  [(cu  +  d)2  +  c2v2]  =  (ad-  be)  v, 

hence  Had  —  be  is  positive,  T?  is  positive  when  v  is  positive.  There  are  three 
effective  constants  in  this  transformation,  namely,  the  ratios  of  a,  b  and  c 
to  d,  hence  by  a  suitable  choice  of  these  constants  any  three  points  on  the 
axis  of  u  may  be  mapped  into  any  three  points  on  the  axis  of  £.  In  fact, 
if  uly  u2,  us  are  the  values  of  u  corresponding  to  the  values  gl9  £2,  £3  °f  l> 
we  can  say  from  the  invariance  of  the  cross-ratio  that 

(£  -  li)  (&  ~  la)  _  (w  ~  ^i)  fa?  -  ^a) 
(£  -  sY)  (&  ~ ~  li)      (^  -^2)  fas  ~  ^i) ' 

and  so  the  equation  of  the  transformation  may  be  written  down  in  the 

previous  form,  the  coefficients  being 

a  =  &&  fa2  ~  ^3)  +  fall  faa  -  %)  4-  Iil2  fai  ~  ^2), 

&  -  ^3)  +  Wa^ila  (la  -  li)  +  ^1^2 Is  (li  ™  I2)» 
|3)  +  ^2  (la  ~  li)  +  ^3  (li  ~  la)» 

-  ^3)  +  7^3^  (|3  -  ^)  +  U1U2  (^  -   f2). 

The  quantity  ad  —  6c  is  given  by  the  formula 
ad  -  be  -  (£2  -  &)  (&  -  ^)  (^  -  f2)  (^2  -  u3)  (u^  -  %)  (^  -  u2). 

If  i^,  u2,  t63  are  all  different  the  coefficients  c  and  d  cannot  vanish 
simultaneously,  for  the  equations  c  =  0,  d  =  0  give 

b2  S3 S3  ""    b  1  _  b  1  ~~    b 2 

«1  ( V  -   %2)  ""  ^2  (^32  ~   %2)  ~  ^3  (^l2  -   ^22)  ' 

and  these  equations  imply  that 

*i  (W22  -  %2)  +  ^2  (^32  ~  %2)  +  ^3  (%2  -  ^22)  =  0, 
or  (w2  -  wj)  (i^  -  Ul)  (HI  -  ^2)  =  0, 

if  the  quantities  fx,  ^2,  ^3  are  also  all  different.  In  a  similar  way  it  can  be 
shown  that  a  and  c  cannot  vanish  simultaneously  and  that  a  and  b  cannot 
vanish  simultaneously.  Poincar6  has  remarked  that  the  transformation 
(II)  can  usually  be  determined  uniquely  so  as  to  satisfy  the  requirement 
that  an  assigned  point  £  and  an  assigned  direction  through  this  point 
should  correspond  to  an  assigned  point  w  and  an  assigned  direction 
through  this  point.  The  proof  of  the  theorem  may  be  left  to  the  reader, 


Riemanrfs  Problem  275 

who  should  examine  also  the  special  case  in  which  one  or  both  of  the 
points  is  on  the  real  axis  in  the  plane  in  which  it  lies. 

EXAMPLES 

1.  Prove  that  Poisson's  formula 

7  =  2*  j  _„.  T-2rcos(6~-'iJ')~-rr*         \r\<l 

maps  into  the  formula  of  §  3*11, 

_  1   f  =°       yF  (x'}  dx' 

~  TT  /  -co  (x  -x')2  +  y*' 
where  /  (0')  =  F  (tan  £0'). 

2.  Prove  that  the  transformation 


maps  the  half  plane  y  >  0  on  the  unit  circle  |  w  |  <  1  in  such  a  way  that  the  point  z0  maps 
into  the  centre  of  the  circle. 

§  4-31.  Riemann's  problem.  The  standard  problem  of  conformal  repre- 
sentation will  be  taken  to  be  that  of  mapping  the  area  A  in  the  z-plane 
.on  the  upper  half  of  the  w-plane  in  such  a  way  that  three  selected  points 
on  the  boundary  of  A  map  into  three  selected  points  on  the  axis  of  u.  This 
is  the  problem  considered  by  B.  Riemann  in  his  dissertation.  The  problem 
may  be  made  more  precise  by  specifying  that  the  function  /  (w)  which 
gives  the  desired  relation 

z=f(w) 

should  possess  the  following  properties : 

(1)  /  (w)  should  be  uniform  and  continuous  for  all  values  of  w  for  which 
v  >  0.   If  WQ  is  any  one  of  these  values/  (w)  should  be  capable  of  expansion 
in  a  Taylor  series  of  ascending  powers  of  w  —  WQ  which  has  a  radius  of 
convergence  different  from  zero. 

(2)  The  derivative/'  (w)  should  exist  and  not  vanish  for  v  >  0 ;  indeed, 
if  /'  (w)  =  0  for  w  =  WQ  there  will  be  at  least  two  points  in  the  neighbour- 
hood of  w0  for  which  z  has  the  same  value.  This  is  contrary  to  the  require- 
ment that  the  representation  should  be  biuniform. 

(3)  /  (?#)  should  be  continuous  also  for  all  real  values  of  tv,  but  it  is 
not  required  that  in  the  neighbourhood  of  one  of  these  values,  WQ,  the 
function  /  (iv)  can  be  expanded  in  a  Taylor  series  of  ascending  powers  of 
w  —  WQ)  for,  as  far  as  the  mapping  is  concerned,  /  (w)  is  defined  only 
for  v  >  0. 

(4)  Considered  as  a  function  of  z,  the  variable  w  should  satisfy  the 
same  conditions  as/  (w).  If/  (w)  satisfies  all  these  requirements  it  will  give 
the  solution  of  the  problem.  The  solution  is,  moreover,  unique  because  if 

two  functions  , ,    .  ,    . 

z  =  f(w),     z  =  g(w) 

give  different  solutions  of  the  problem,  the  transformation 

18-2 


276  Conformal  Representation 

will  map  the  upper  half  plane  into  itself  in  such  a  way  that  the  points 
^i >  uz )  U3  m&p  into  themselves.  Now  it  can  be  proved  that  a  transformation 
which  maps  the  upper  half  plane  into  itself  is  bilinear  and  so  the  relation 
between  w  and  W  is 

(w  -  uj  (u2  -  u3)  ^  (W  -7^)  (w_2_-  ?/3) 

(w  -  u2)  (u3  -  uj '    (W  -  u2)  (w.a  -  u^ 

w  —  HI      W  —  u, 

mr  A  A 

ui  — —  . —   .  . 

W  —  U2         W  —  U2 

This  reduces  to 

(11,  -  W)  (u,  -  u2)  -  0, 
and  so  W  =  w. 

§  4-32.  TAe  (jeneral  problem  of  conformed  representation.  The  general  type 
of  region  which  is  considered  in  the  theory  of  conformal  representation 
may  be  regarded  as  a  carpet  which  is  laid  down  on  the  z -plane.  This  carpet 
is  supposed  to  have  a  boundary  the  exact  nature  of  which  requires  careful 
specification  because  with  the  aim  of  obtaining  the  greatest  possible 
generality,  different  writers  use  different  definitions  of  the  boundary  curve. 
There  may,  indeed,  be  more  than  one  boundary  curve,  for  a  carpet  may, 
for  instance,  have  a  hole  in  its  centre.  For  simplicity  we  shall  suppose 
that  each  boundary  curve  is  a  simple  closed  curve  composed  of  a  finite 
number  of  pieces,  each  piece  having  a  definite  direction  at  each  of  its 
points.  At  a  point  where  two  pieces  meet,  however,  the  directions  of  the 
two  tangents  need  not  be  the  same;  a  carpet  may,  for  instance,  have  a 
corner.  The  tangent  may  actually  turn  through  an  angle  2?r  as  we  pass 
from  one  piece  of  a  boundary  curve  to  another  and  in  this  case  the  boundary 
has  a  sharp  point  which  may  point  either  inwards  or  outwards.  It  turns 
out  that  the  fojmer  case  presents  a  greater  difficulty  than  the  latter. 

In  special  investigations  other  restrictions  may  be  laid  on  each  piece 
of  a  boundary  curve  and  from  the  numerous  restrictions  which  have  beer 
used  we  shall  select  the  following  for  special  mention. 

(1)  The  direction  of  the  tangent  is  required  to  vary  continuously  as 
a  point  moves  along  the  curve  (smooth  curve*). 

(2)  The  curvature  is  required  to  vary  continuously  as  a  point  moves 
along  the  curve. 

(3)  The  curve  should  be  rectifiable,  i.e.  it  should  be  possible  to  define 
the  length  of  any  portion  of  the  curve  with  the  aid  of  a  definite  integral 
which  has  a  precise  meaning  specified  beforehand,  such  as  the  meaning 
given  to  an  integral  by  Riemann,  Stieltjes  or  Lebesgue. 

A  simple  curve  which  possesses  the  first  property  may  be  called  a 
curve  (CT),  one  which  possesses  the  properties  1  and  2  a  curve  (CTC), 
a  curve  which  possesses  the  properties  1  and  3  may  be  called  a  curve  (RCT). 

*  A  curve  which  is  made  up  of  pieces  of  smooth  curves  joined  together  may  be  called  smooth 
bit  by  bit  ("Stiickweise  glatte  Kurve";  see  Hurwitz-Courant,  Berlin  (1925),  Funktionentheorie). 


Properties  of  Regions  277 

The  carpet  will  be  said  to  be  simply  connected  when  a  cross  cut  starting 
from  any  point  of  the  boundary  and  ending  at  any  other  divides  the  carpet 
into  two  pieces.  A  carpet  shaped  like  a  ring  is  not  simply  connected 
because  a  cut  starting  from  a  point  on  one  boundary  and  ending  at  a  point 
on  the  other  does  not  divide  the  carpet  into  two  pieces.  ^Such  a  carpet 
may,  however,  be  made  simply  connected  by  making  a  cut  of  this  type. 
When  we  consider  a  carpet  with  n  boundaries  which  are  simple  closed 
curves  we  shall  suppose  that  the  boundaries  can  be  converted  into  one 
by  a  suitable  number  of  cuts  which  will  at  the  same  time  render  the  carpet 
simply  connected.  It  will  be  supposed,  in  fact,  that  the  carpet  is  not 
twisted  like  a  Mobius'  strip  when  the  cuts  have  been  made. 

Any  closed  curve  on  a  simply  connected  carpet  can  be  continuously 
deformed  until  it  becomes  an  infinitely  small  circle.  This  cannot  always  be 
done  on  a  ring-shaped  carpet  as  may  be  seen  by  considering  a  circle  con- 
centric with  the  boundary  circles  of  a  ring,  and  it  cannot  be  done  in  the 
case  of  a  curve  which  runs  parallel  to  the  edge  of  a  singly  twisted  Mobius' 
strip  formed  by  joining  the  ends  of  a  thin  rectangular  strip  of  paper  after 
the  strip  has  been  given  a  single  twist  through  180°.  Such  a  closed  curve 
is  said  to  be  irreducible  and  the  connectivity  of  a  carpet  may  be  defined 
with  the  aid  of  the  number  of  different  types  of  irreducible  closed  curves 
that  can  be  drawn  on  it.  Two  closed  curves  are  said  to  be  of  different  types 
when  one  cannot  be  deformed  into  the  other  without  any  break  or  crossing 
of  the  boundary.  It  is  not  allowed,  for  instance,  to  cut  the  curve  into 
pieces  and  join  these  together  later  or  in  any  way  to  make  the  curve  into 
one  which  does  not  close. 

A  simply  connected  carpet  may  cover  the  plane  more  than  once;  it 
may,  for  instance,  be  folded  over,  or  it  may  be  double,  triple,  etc.  In  the 
latter  case  it  is  called  a  Riemann  surface,  i.e.  a  surface  consisting  of  several 
sheets  which  connect  with  one  another  at  certain  branch  lines  in  such  a 
way  as  to  give  a  simply  connected  surface.  When  there  are  only  two 
sheets  it  is  often  convenient  to  regard  them  as  the  upper  and  lower 
surfaces  of  a  single  carpet  with  a  cut  or  branch  line  through  which  passage 
may  be  made  from  one  surface  to  the  other.  In  the  case  of  a  ring-shaped 
carpet  we  generally  consider  only  the  upper  surface,  but  if  the  lower  surface 
is  also  considered  and  a  passage  is  allowed  from  one  surface  to  the  other 
across  either  one  or  both  of  the  edges  of  the  ring  a  surface  with  two  sheets 
is  obtained,  but  this  doubly  sheeted  surface  is  not  simply  connected 
because  a  curve  concentric  with  the  two  edges  is  still  irreducible.  In  a  more 
general  theory  of  conform al  representation  the  mapping  of  multiply 
connected  siirf  aces  is  considered,  but  these  will  be  excluded  from  the  present 
considerations . 

A  carpet  may  also  have  an  infinite  number  of  boundaries  or  an  infinite 
number  of  sheets,  but  these  cases  will  also  be  excluded.  When  we  speak  of 


278  Conformal  Representation 

an  area  A  we  shall  mean  the  right  side  of  a  carpet  which  is  bounded  by  a 
simple  closed  curve  formed  of  pieces  of  type  (RCTC)  and  is  not  folded 
over  in  any  way.  The  carpet  will  be  supposed,  in  fact,  to  be  simply  con- 
nected and  smooth,  the  word  smooth  being  used  here  as  equivalent  to  the 
German  word  "schlicht,"  which  means  that  the  carpet  is  not  folded  or 
wrinkled  in  any  way.  The  function  F  (z)  maps  the  circular  area  |  z  \  <  1 
into  a  smooth  region  if 

F  fa)  -  F  (z2) 

£-  1—  *  °' 

whenever  |  zl  \  <  1  and  |  z2  |  <  1  . 

We  shall  be  occupied  in  general  with  the  conformal  representation  of 
one  simple  area  on  another,  and  for  brevity  we  shall  speak  of  this  as  a 
mapping.  In  advanced  works  on  the  theory  of  functions  the  problem  of 
conformal  representation  is  considered  also  for  the  case  of  Riemann 
surfaces  and  the  more  general  theory  of  the  conformal  representation  of 
multiply  connected  surfaces  is  treated  in  books  on  the  differential  geometry 
of  surfaces. 

For  many  purposes  it  will  be  sufficient  to  consider  the  problem  of 
conformal  representation  for  the  case  of  boundaries  made  up  of  pieces  of 
curves  having  the  property  that  the  co-ordinates  of  their  points  can  be 
expressed  parametrically  in  the  form 


where  the  functions  /  (t)  and  g  (t)  can  be  expanded  in  power  series  of  type 

2an(*-*0)n,  ......  (HI) 

n-O 

which  are  absolutely  and  uniformly  convergent  for  all  values  of  the 
parameter  t  that  are  needed  for  the  specification  of  points  on  the  arc  under 
consideration.  In  such  a  case  the  boundary  is  said  to  be  composed  of 
analytic  curves  and  this  is  the  type  of  boundary  that  was  considered  in 
the  pioneer  work  of  H.  A.  Schwarz,  but  the  restriction  of  the  theory  to 
boundaries  composed  of  analytic  curves  is  not  necessary*  and  a  method 
of  removing  this  restriction  was  found  by  W.  F.  Osgoodf.  His  work  has 
been  followed  up  by  that  of  many  other  investigators^. 

In  the  power  series  (III)  the  quantities  tQ,  an  are  constants  which,  of 
course,  may  be  different  for  different  pieces  of  the  boundary. 

There  are  really  two  problems  of  conformal  representation.  In  one 
problem  the  aim  is  simply  to  map  the  open  region  A  bounded  by  a  curve 

*  In  the  modern  work  the  boundary  considered  is  a  Jordan  curve,  that  is,  a  curve  whose  points 
may  be  placed  in  a  continuous  (1,1)  correspondence  with  the  points  of  a  circle. 
t  Trans.  Amer.  Math.  Soc.  vol.  i,  p.  310  (1900). 
t  Particularly  E.  Study,  C.  Caratheodory,  P.  Koebe  and  L.  Bieberbach. 


Exceptional  Cases  279 

a  on  the  open  region  B  bounded  by  a  curve  6,  there  being  no  specified 
requirement  about  the  correspondence  of  points  on  the  two  boundaries. 
In  the  second  problem  the  aim  is  to  map  the  closed  realm*  A  on  the 
closed  realm  B  in  such  a  way  that  each  point  P  on  a  corresponds  to  only 
one  point  Q  which  is  on  b  and  so  that  each  point  Q  on  b  corresponds  to 
only  one  point  P  which  is  on  a.  It  is  this  second  problem  which  is  of  most 
interest  in  applied  mathematics.  If,  moreover,  in  the  first  problem  the 
correspondence  between  the  boundaries  is  not  one-to-one  the  applied 
mathematician  is  anxious  to  know  where  the  uniformity  of  the  corre- 
spondence breaks  down. 

Existence  theorems  are  more  easily  established  for  the  first  problem 
than  for  the  second  and  fortunately  it  always  happens  in  practice  that 
a  solution  of  the  first  problem  is  also  a  solution  of  the  second ;  but  this,  of 
course,  requires  proof  and  such  a  proof  must  be  added  to  an  existence 
theorem  that  is  adapted  only  for  the  first  problem. 

The  methods  of  conf ormal  representation  are  particularly  useful  because 
they  frequently  enable  us  to  deduce  the  solution  of  a  boundary  problem 
for  one  closed  region  A  from  the  solution  of  a  corresponding  boundary 
problem  for  another  region  B  which  is  of  a  simpler  type.  When  the  function 
which  effects  the  mapping  is  given  by  an  explicit  relation  the  process  of 
solution  is  generally  one  of  simple  substitution  of  expressions  in  a  formula, 
but  when  the  relation  is  of  an  implicit  nature  or  is  expressed  by  an  infinite 
series  or  a  definite  integral  the  direct  method  of  substitution  becomes 
difficult  and  a  method  of  approximation  may  be  preferable.  A  method  of 
approximation  which  is  admirable  for  the  purpose  of  establishing  the 
existence  of  a  solution  may  not  be  the  best  for  purposes  of  computation. 

§  4-33.  Special  and  exceptional  cases.  It  is  easy  to  see  that  it  is  not 
possible  to  map  the  whole  of  the  complex  z-plane  on  the  interior  of  a  circle. 
Indeed,  if  there  were  a  mapping  function  /  (z)  which  gave  the  desired 
representation,  /  (z)  would  be  analytic  over  the  whole  plane  and  |  /  (z)  \ 
would  always  lie  below  a  certain  positive  value  determined  by  the  radius  of 
the  circle  into  which  the  z-plane  maps,  consequently  by  Liouville's  theorem 
/  (z)  would  be  a  constant.  A  similar  argument  may  be  used  for  the  case 
of  the  pierced  z-plane  with  the  point  z0  as  boundary.  By  means  of  a 
transformation  z  —  z0  =  1/z'  the  region  outside  z0  can  be  mapped  into  the 
whole  of  the  z'-plane  when  the  point  z'  =  oo  is  excluded.  The  mapping 
function  is  again  an  integral  function  for  which  \f(z')\<M,  and  is  thus 
a  constant.  On  account  of  this  result  a  region  considered  in  the  mapping 
problem  is  supposed  to  have  more  than  one  external  point,  a  point  on  the 
boundary  being  regarded  as  an  external  point. 

*  We  use  realm  as  equivalent  to  the  German  word  "  Bereich, "  and  region  as  equivalent  to 
"  Gebiet." 


280  Conformal  Representation 

The  next  case  in  order  of  simplicity  is  the  simply  connected  region 
with  at  least  two  boundary  points  A  and  B.  If  these  were  isolated  the 
region  would  not  be  simply  connected.  We  shall  therefore  assume  that 
there  is  a  curve  of  boundary  points  joining  A  and  B.  This  curve  may 
contain  all  the  boundary  points  (Case  1)  or  it  may  be  part  of  a  curve  of 
boundary  points  which  may  either  be  closed  or  terminated  by  two  other 
end-points  C  and  F.  The  latter  case  is  similar  to  the  first,  while  the  case 
of  a  closed  curve  is  the  one  which  we  wish  eventually  to  consider. 

The  simplest  example  of  the  first  case  is  that  in  which  the  end-points 
are  z  -=  0  and  z  =  oo,  the  boundary  consisting  of  the  positive  x-axis.  The 
region  bounded  by  this  line  can  be  regarded  as  one  sheet  of  a  two-sheeted 
Riemann  surface  with  the  points  0  and  oo  as  winding  points  of  the  second 
order,  passage  from  one  sheet  to  the  other  being  made  possible  by  a 
junction  of  the  sheets  along  the  positive  #-axis.  The  whole  of  this  Riemann 
surface  is  mapped  on  the  z' -plane  by  means  of  the  simple  transformation 
z'  —  y'z,  which  sends  the  one  sheet  in  which  we  are  interested  into  the 
half  plane  0  <  6'  <  77,  where  z'  =  r'elQf. 

In  the  case  when  the  boundary  consists  of  a  curve  joining  the  points 
z  =  a,  z  =•-  6,  these  points  are  regarded  as  winding  points  of  the  second 
order  for  a  two-sheeted  Riemann  surface  whose  sheets  connect  with  each 
other  along  the  boundary  curve.  This  surface  is  mapped  on  the  whole 
z'-plane  by  means  of  the  transformation 


'  (Z  ~  "}*      cz'  ~  1 
(z  -~b)    -  cz   -  *> 


and  in  this  transformation  one  sheet  goes -into  the  interior,  the  other  into 
the  exterior  of  a  certain  closed  curve  (7.  The  mapping  problem  is  thus 
reduced  to  the  mapping  of  the  interior  of  C  on  a  half  plane  or  a  unit  circle. 
Finally,  by  means  of  a  transformation  of  type 

z=  Az'  +  B, 

we  can  transform  the  region  enclosed  by  C  into  a  region  which  lies  entirely 
within  the  unit  circle  |  z  \  <  1  and  our  problem  is  to  map  this  region  on 
the  interior  of  the  unit  circle  |  £  |  <  1  by  means  of  a  transformation  of 
type  J=/(z). 

§  4-41.  The  mapping  of  the  unit  circle  on  itself.  If  a  and  a  are  any  two 
conjugate  complex  quantities  and  a  is  a  real  angle  the  quantities  e*a  —  a 
and  1  —  dela  have  the  same  modulus,  consequently  if  )3  is  another  real 
angle  the  transformation 

(1  -&)  £  =  e*(z-a)  (A) 

maps  |  z  |  =  1  into  |  £  |  =  1,  and  it  is  readily  seen  that  the  interior  of  one 
circle  maps  into  the  interior  of  the  other.  It  should  be  noticed  that  this 


Circle  Mapped  on  Itself  281 

transformation  maps  the  point  z  =  a  into  the  centre  of  the  circle  |  £  |  =  1, 
If  we  put  a  =  0  the  transformation  reduces  to  the  rotation 

£  =  ze*, 

which  leaves  the  centre  of  the  circle  unaltered.  If  we  can  prove  that  this 
is  the  most  general  conformal  transformation  which  maps  the  interior  of 
the  unit  circle  into  itself  in  such  a  way  that  the  centre  maps  into  the 
centre  it  will  follow  that  the  formula  (A)  gives  the  most  general  trans- 
formation which  maps  the  unit  circle  into  itself. 

The  following  proof  is  due  to  H.  A.  Schwarz. 

Let  /  (z)  be  an  analytic  function  of  z  which  is  regular  in  the  circle 
|  z  |  =  1  and  satisfies  the  conditions 

\f(z)\<  1   for   \z\<  1,    /(0)  =  0. 

If  "<£(*)  =/(*)/*>     </>(<>)  =/'(0), 

the  function  </>  (z)  is  also  regular  in  the  unit  circle,  and  if  |  z  \  =  r,  where 
r  <  I,1  we  have 

I  <£  (z)  I  <  l/r- 

But  since  </>  (z)  is  analytic  in  the  circle  |  z  \  =  r  the  maximum  value 
of  |  <f>  (z)  |  occurs  on  the  boundary  of  this  region  and  not  within  it,  hence 
for  a  point  z0  within  the  circle  |  z  \  =  r,  or  on  its  circumference,  we  have 
the  inequality  |  <f>  (z0)  |  <  l/r  (Schwarz's  inequality*). 

Passing  to  the  limit  r  ->  1  we  have  the  inequality 
|  <£  (z0)  |  <  1  for  |  z0  |  <  1. 

Now  let  £  =  /  (z),  z  -=  g  (£)  be  the  mapping  functions  which  map  a 
circle  on  itself  in  such  a  way  that  the  centre  maps  into  the  centre,  then 
by  Schwarz's  inequality 

|  £/z  |  <  1,      |  z/£  |  <  1. 

Rence  |  £/2  |  =•=  1,  and  so  |  </>  (z)  \  is  equal  to  unity  within  the  unit 
circle.  Now  an  analytic  function  whose  modulus  is  constant  within  the 
unit  circle  is  necessarily  a  constant,  hence  £  =  zel?  where  j8  is  a  constant 
real  angle. 

Since  the  unit  circle  is  mapped  on  a  half  plane  by  a  bilinear  trans- 
formation, it  follows  that  a  transformation  which  maps  a  half  plane  into 
itself  is  necessarily  a  bilinear  transformation. 

§  4-42.  Normalisation  of  the  mapping  problem.  Let  F  be  the  unit  circle 
|  £  |  <  1  in  the  £-plane  and  suppose  that  a  smooth  region  G  in  the 
z-plane  can  be  mapped  in  a  (1,  1)  manner  on  the  interior  of  F.  Since  F 
can  be  mapped  on  itself  by  a  bilinear  transformation  in  such  a  way  that 
two  prescribed  linear  elements  correspond,  it  is  always  possible  to  normalise 

*  This  is  often  called  Schwarz's  lemma  as  another  inequality  is  known  as  Schwarz's  inequality. 
The  lemma  of  §  4*61  is  then  called  Schwarz's  principle  or  continuation  theorem.  This  second  in- 
equality is  used  in  §  4-81. 


282  Conformal  Representation 

the  mapping  so  that  a  prescribed  linear  element  in  the  region  0  corresponds 
to  the  centre  of  the  unit  circle  and  the  direction  of  the  positive  real  axis, 
that  is  to  what  we  may  call  the  "  chief  linear  element."  We  can  then, 
without  loss  of  generality,  imagine  the  axes  in  the  z-plane  to  be  chosen  so 
that  the  origin  lies  in  0  on  the  prescribed  linear  element  and  so  that  this 
linear  element  is  the  chief  linear  element  for  the  z-plane.  This  means  that 
the  normalised  mapping  function  £  =  /  (z)  satisfies  the  conditions/  (0)  =  0, 
/'  (0)  >  0.  Finally,  by  a  suitable  choice  of  the  unit  of  length  in  the  z-plane, 
or  by  a  transformation  of  type  z'  =  kz,  we  can  make/'  (0)  =  1.  The  trans- 
formation is  then  fully  normalised  and  /  (z)  is  a  completely  normalised 
mapping  function.  The  power  series  which  represents  the  function  in  the 
neighbourhood  of  z  =  0  is  of  type 

/(z)  -  z-f  a2z2+  .... 

The  coefficients  a2>  as>  •••  *n  this  series  are  not  entirely  arbitrary,  in 
fact  it  appears  that  a2  is  subject  to  the  inequality*  |  a2  \  <  2.  To  prove  this 
we  consider  the  function  g  (z)  defined  by  the  equation  /  (z)  g  (z)  =  1  .  We 


If  0<c<  1,  the  transformation  y  =  g  (z)  maps  the  circular  ring 
c  <  z  <  1  on  a  region  A  in  the  y-plane  bounded  by  a  curve  C  and  a  curve 
Cc  which  can  be  represented  parametrically  by  the  equation 

cy  =  e-«*  -f  6xc  +  &2cV*  +  c3o>  (c,  a), 

where  a  is  the  parameter  and  co  (c,  a)  remains  bounded  as  a  varies  between 
0  and  277.   Writing  62  =  be2*,  cd  =  1,  we  remark  that  the  equation 


^1  V 

cJO 


gives  the  parametric  representation  of  an  ellipse  with  semi-axes  d  +  be, 
d  —  be  respectively,  and  so  the  area  Ac  of  the  curve  Ce  differs  from  nd2 
by  cB,  where  |  B  \  remains  bounded  as  c  -*  0.  Now  the  area  of  the  region  A 
is  a  quantity  Ac'  given  by  the  equation 

-l+   S  n  |  bn  I2  -    S  n  \  bn  I2  c 

n-2  n-2 

and  Ac  >  Ac'-,  also  as  c  -*>  0  the  difference  nd*  —  Ac'  tends  to  the  area  A 
of  the  region  enclosed  by  C.  Since  A  >  0  we  have  the  inequality 

S  n\bn\*<\. 

n-2 

Now  the  function 

[<7  (z2)]*  =  ^  -f  ftz+...,     20!=^ 

likewise  maps  the  unit  circle  |  z  |  <  1  on  a  smooth  region,  and  so  by  the 

last  theorem  ,  Q  , 

\  Pi\    <  !• 
*  See,  for  instance,  L.  Bieberbach,  Berlin.  Sitzungsber.  Bd.  xxxvm,  S.  940  (1916). 


Inequalities  283 

Since  bl=  —  a2  the  last  inequality  becomes  simply 
|  «2  |2<  4   or    |  a2  |  <  2. 

We  have  |  &  |  =  1  when  and  only  when  /?2  =  /?3  =  ...  =  0,  consequently 
|  a2  |  =  2  when  and  only  when 

[g  (z2)]^  =  -  -f  zeicr,  where  a  is  real, 

z 

or  gr(z)  =  (z~*  +  eM)2, 

that  is,  when  /  (z)  =  (  L  -*^  . 

EXAMPLES 

1.  A  transformation  which  maps  the  unit  circle  into  itself  in  a  one-to-one  manner  and 
transforms  the  chief  linear  element  into  itself  is  necessarily  the  identical  transformation. 

[Schwarz,  and  Poincare".] 

2.  A  region  enclosing  the  origin  which  can  be  mapped  on  itself  with  conservation  of  the 
chief  linear  element  consists  either  of  the  whole  plane  or  of  the  whole  plane  pierced  at  the 
origin.  [T.  Rad6,  Szeged  Acta,  1.  1,  p.  240  (1923).] 

3.  If  the  region  |  z  \  <  1  is  mapped  smoothly  on  a  region  W  in  the  w-  plane  by  the  function 
w  =f(z)  =  z  +  ajz2  +  agZ3  -f-  ...,  prove  that,  when  |  z  \  =  r  <  1, 


Hence  show  that  if  WQ  is  a  point  not  belonging  to  the  region  W 

I  wo  I  <  i- 
The  value  |  w0  \  =  J  is  attained  at  the  point  w0  =  Je"*"  when 


4.  If  a  region  W  of  the  w-plane  is  mapped  smoothly  on  the  circle  |  z  \  <  1  by  the  f  unction 
w  =/(z)  and  if  Z19  Z2  are  any  two  points  which  do  not  lie  within  the  circle,  then 


[G.  Pick,  Leipziger  Berichte,  Bd.  LXXXJ,  S.  3  (1929).] 

§  4-43.  The  derivative  of  a  normalised  mapping  function.  Now  let  /  (z) 
be  regular  in  the  unit  circle  |  z  |  <  1,  which  we  shall  call  K.  We  shall  study 
the  behaviour  of  /'  (z)  in  the  neighbourhood  of  an  interior  point  z0  of  K. 
Let  z0  be  the  conjugate  of  z0,  then  the  transformation 

z'  -  Z  ~  *° 

1         ZZ0 

maps  the  circle  K  into  itself  and  sends  the  point  z0  into  the  point  z'  =  0. 
Thus 


284 


Conformal  Representation 

/*(z)=/(iz+-!°2)-/(z<>)> 


Writing 

we  see  that  the  function  /*  (z)  maps  the  circle  K  on  a  smooth  region  and 
leaves  the  origin  fixed.   If  z0  =  reie  we  have  by  Taylor's  theorem 

/*  (z)  -  z  (1  -  r»)/'  (z0)  +  ¥2  (1  -  ^)  [(1  -  r2)  /"  fo)  -  2z0/'  (z0)]  +  -  • 
The  function! 

*M-<.^ta>-'  +  ^1+"- 

is  thus  a  normalised  mapping  function,  and  so  by  the  theorem  of  §  4-42, 
I  A2  |  <  2,  i.e. 

'(1 


Writing  z0J     v;o;  -  P  +  iQ  -  r  ^  (w  -1-  iv), 

where/'  (2)  —  eu+tv  and  ^  and  v  are  real,  we  have  the  inequality 

4r 


Therefore 


4  -  2r 


:  ]_ , 

4  -f  2r 

4 


4          3v 
""  1  -  ra<  3r  ^  1  -  r2" 

Integrating  between  0  and  r  we  obtain  the  twa  inequalities  J 

1  -  r     ^       ^  ,         I  -f  r 


log 


( 1  ~ 


rFhe  first  of  these  may  be  written  in  the  more  general  form 


. 

(1+  |z|)'~ 


/'  (z) 
/'(O) 


1   I- 
(1  - 


where  now  ^  =•  f  (z)  is  a  function  which  maps  K  on  a  smooth  region  not 

f  This  function  was  used  by  L.  Bicberbach,  Math.  Zeitschr.  Bd.  iv,  S.  295  (1919),  and  later  by 
R.  Nevanlinna,  see  Bieberbach,  Math.  Zeitechr.  Bd.  ix,  8.  161  (1921).  The  following  analysis 
which  is  due  to  Nevanlinna  is  derived  from  the  account  in  Hurwitz-Courant,  Funktionentheone, 
S.  388  (1925). 

t  The  first  of  these  was  given  by  T.  H.  Gronwall,  Comptes  Rendus,  t.  CLXII,  p.  249  (1916), 
and  by  J.  Plemelj  and  G.  Pick,  Leipziger  Ber.  Bd.  LXVIII,  S.  58  (1916).  In  the  form 

k(r)<\f'(z)\  <h(r)  • 

it  is  known  as  Koebe's  Verzerrungssatz  (distortion  theorem).  The  precise  forms  for  k  (r)  and  h  (r) 
were  derived  also  by  G.  Faber,  Munchener  Ber.  S.  39  (1916)  and  L.  Bieberbach,  Berliner  Ber. 
Bd.  xxxvm,  S.  940  (1916).  The  inequality  satisfied  by  |  v  \  was  discovered  by  Bieberbach,  Math. 
Zeitschr.  Bd.  iv,  S.  295  (1919). 


Properties  of  the  Derivative  285 

containing  the  point  at  infinity  but  is  not  necessarily  a  normalised  mapping 
function. 

The  second  theorem  is  called  the  rotation  theorem,  as  it  indicates  limits 
for  the  angle  through  which  a  small  area  is  rotated  in  the  conformal 
mapping.  The  other  theorem  gives  limits  for  the  ratio  in  which  the  area 
changes  in  size.  This  theorem  has  been  much  used  by  Koebe*  in  his 
investigations  relating  to  the  conformal  representation  of  regions  and  has 
been  used  also  in  hydrodynamics^  and  aerodynamics. 

When  /  (z)  maps  K  on  a  convex  region  it  can  be  shown  that  |  /'  (z)  \ 
lies  within  narrower  limitsj.  Study  has  shown,  moreover,  that  in  this  case 
any  circle  within  K  and  concentric  with  it  also  maps  into  a  convex  region§. 
Many  other  inequalities  relating  to  conformal  mapping  are  given  in  a  paper 
by  J.  E.  Littlewood,  Proc.  London  Math.  Soc.  (2),  vol.  xxm,  p.  481  (1925). 

§  4-44.  The  mapping  of  a  doubly  carpeted  circle  with  one  interior  branch 
point.  Let  P  be  a  point  within  the  unit  circle  |  z'  \  <  1  and  let  r2eld 
(0  <  r  <  1)  be  the  value  of  z'  at  P.  The  transformation || 

(1  +  r2)z-  2re*       .  ,  x  ,.. 

z  =  *2£— (i^r'je*'^  (A) 

satisfies  the  conditions  </>  (0)  =  0,  </>'  (0)  >  0,  |  z'  \  =  \  z  j,  when  z  \  =-  1,  and 
so  represents  a  partially  normalised  transformation  which  maps  the  unit- 
circle  in  the  z-plane  on  a  doubly  carpeted  unit  circle  in  the  z'-plane,  the 
two  sheets  having  a  junction  along  a  line  extending  from  P  to  the  boundary. 
We  shall  regard  this  line  as  a  cut  in  that  sheet  which  contains  the  chief 
element  corresponding  to  the  chief  element  in  the  z-plane. 

It  is  evident  that  |  z'  \  <  \  z  \  whenever  |  z  \  <  1 ,  and  so  |  z  \  <  1  when 
|  z  |  <  1.  This  means  that  |  z  \  <  \z\  whenever  |  z'  \  <  1. 

From  this  we  conclude  that  for  all  values  of  z'  for  which  |  z'  \  <  r2  there 
is  a  positive  number  q  (r)  greater  than  unity  for  which  |  z  \  >  q  (r)  \  z'  \ . 
Indeed,  if  there  were  no  such  quantity  g  (r)  there  would  be  at  least  one 
point  in  the  circle  |  z'  \  <  r2,  for  which  z  \  =  |  z'  \ .  An  expression  for  q  (r) 
may  be  obtained  by  writing 

2r 

'-l  +  r"     *  =  ?*•. 

and  considering  the  points  s,  l/s  on  the  real  axis.  If  El ,  R2  are  the  distances 
of  the  point  z  from  these  points  respectively  we  have 

I 'I- -at 

*  Gott.  Nachr.  (1909);  Crdle,  Bd.  cxxxvm,  S.  248  (1910);  Math.  Ann.  Bd.  LXIX. 

f  Ph.  Frank  and  K.  Lowner,  Math.  Zeitschr.  Bd.  in,  S.  78  (1919). 

t  T.  H.  Gronwall,  Comptes  Rendus,  t.  CLXII,  p.  316  (1916). 

§  E.  Study,  Konforme  Abbildung  einfazh  zusammenhiingender  Bereiche,  p.  110  (Teubner, 
Leipzig,  1913).  A  simple  proof  depending  on  a  use  of  Schwarz's  inequality  has  been  given  recently 
by  T.  Rado,  Math.  Ann.  Bd.  en,  S.  428  (1929). 

H  C.  Caratheodory,  Math.  Ann.  Bd.  Lxxn,  S.  107  (1912). 


286  Conformal  Representation 

The  oval  curve  for  which  pR1/sR2  =  r2  lies  entirely  within  a  circle 
R2/R1  =  constant  which  touches  it  at  a  point  #0  =  —  p  on  the  real  axis 
for  which  /  ,  \  2 


ps), 


r 


l  +  r,  [(2  +  2r*)i  -  1  +  r«]. 


The  constant  is  found  to  be 


and  we  may  take  this  as  our  value  of  q  (r).   We  can  see  that  it  is  greater 
than  1  when  r  <  1  because 

2  f  2r4  -  (1  -f  r  -  r2  +  r3)2  -  (1  -  r4)  (1  -  r)2. 

It  is  clear  from  the  inequality  |  z  \  >  q  (r)  \  z'  \  that  points  corre- 
sponding to  those  which  lie  within  the  circle  |  z  \  <  r2  in  the  z'  -plane  lie 
within  the  larger  circle  |  z  |  <  r2q  (r)  in  the  z-plane.  If  r2  is  the  minimum 
distance  from  the  origin  of  points  on  a  closed  continuous  curve  C"  which 
lies  entirely  within  the  unit  circle  |  *'  |  <  1,  the  transformation  (A)  maps 
the  interior  of  C'  into  the  interior  of  a  closed*  curve  C  which  lies  entirely 
between  the  two  circles  |  z  \  <  1,  |  z  \  =  r2q  (r).  The  shortest  distance  "from 
the  origin  to  a  point  of  C  may  be  greater  than  r2q  (r)  but  it  lies  between 
this  quantity  and  r,  i.e.  the  value  of  |  z  |  corresponding  to  the  branch 
point  z  =  eier2.  This  second  minimum  distance  may  be  used  as  the  constant 
of  type  r2  in  a  second  transformation  of  type  (A).  Let  us  call  it  rx2  and 
use  the  symbol  Cl  to  denote  the  curve  into  which  C  is  mapped  by  the  new 
transformation.  The  minimum  distance  from  the  origin  of  a  point  of  this 
curve  is  a  quantity  r22  which  is  not  less  than  a  quantity  r-^q  (rx)  associated 
with  the  number  rx  . 

If  we  consider  the  worst  possible  case  in  which  the  minimum  distance 
for  a  curve  Cn+l  derived  from  a  curve  Cn  with  minimum  distance  rn2  is 
always  rnzq  (rn),  we  have  a  sequence  of  numbers  rl9  r2,  ...  rn,  which  -are 

derived  successively  by  means  of  the  recurrence  relation 

• 

W  =  ,-^V,  [(2  +  2r.«)*  -  1  +  rn»]. 

i  ~r  rn  » 

Since  rn  <  1  for  all  values  of  n  and  rn4l>  rn,  the  sequence  tends  to 
a  limit  R  which  must  be  given  by  the  equation 

R*  =  -Jj-fr  [(2  +  25*)*  -  1  +  R*l 

This  equation  gives  the  value  R  =  1.  Hence  as  n  ->oo  the  curve  Cn  lies 
between  two  circles  which  ultimately  coincide. 

*  The  curve  C  may  in  some  cases  close  by  crossing  the  line  which  corresponds  to  the  cut  in 
the  z'-plane.  This  will  not  happen  if  the  cut  is  drawn  so  that  it  does  not  intersect  C'  again. 


Sequence  of  Mapping  Operations  287 

The  convergence  to  the  limit  is  very  slow,  as  may  be  seen  by  considering 
a  few  successive  values  of  rn2 : 

r  2  r  2  r  2 

rl  r2  r3 

•25      -283      -309 

The  best  possible  case  from  the  point  of  view  of  convergence  is  that  in 
which  /i2  =  r.  This  case  occurs  when  the  curve  C"  is  shaped  something  like 
a  cardioid  with  a  cusp  at  P. 

Though  useful  for  establishing  the  existence  of  the  conformal  mapping 
of  a  region  on  the  unit  circle,  the  present  transformation  is  not  as  useful 
as  some  others  for  the  purpose  of  transforming  a  given  curve  into  another 
curve  which  is  nearly  circular,  unless  the  given  curve  happens  to  be  shaped 
something  like  a  cardioid  or  a  Iima9on  with  imaginary  tangents  at  the 
double  point.  We  shall  not  complete  the  proof  of  the  existence  of  a 
mapping  function  for  a  region  bounded  by  a  Jordan  curve.  This  is  done 
in  books  on  the  theory  of  functions  such  as  those  of  Bieberbach  and 
Hurwitz-Courant.  Reference  may  be  made  also  to  the  tract  on  conformal 
transformation  which  is  being  written  by  Caratheodory*,  to  E.  Goursat's 
Cours  d' Analyse  Mathematique,  t.  in,  and  to  Picard's  Traite  d' Analyse. 

§  4-45.  The  selection  theorem.  Let  us  suppose  that  the  set  or  sequence 
of  functions  u±  (x,  y),  u2  (x,  y),  u3  (x,  y),  ...  possesses  the  following  pro- 
perties : 

(1)  It  is  uniformly  bounded.  This  means  that  in  the  region  of  definition 
E  the  functions  all  satisfy  an  inequality  of  type 

I  u8  (x,  y)  |  <  M , 

where  M  is  a  number  independent  of  s  and  of  the  position  of  the  point 
(x,  y)  of  the  region  R. 

(2)  It  is  equicontinuous^ .  This  means  that  for  any  small  positive  number 
c  there  is  an  associated  number  S  independent  of  s,  x  and  y  but  depending 
on  €  in  such  a  way  that  whenever 

(x'  -  xY  +  (yr  -  y)2  <  S2 
we  have  |  u8  (x' ',  y')  -  us  (x,  y)  \  <  \c. 

We  now  suppose  that  the  sequence  contains  an  unlimited  number  of  func- 
tions and  that  an  infinite  number  of  these  functions  forming  a  subsequence 
^wi  (x>  y)>  Um2  (x>  y}>  •••  can  be  selected  by  a  selection  rule  (m).  Our  aim 
now  is  to  find  a  sequence  (1),  (2),  (3),  ...  of  selection  rules  such  that  the 
"diagonal  sequence "  un  (x,  y),  u22  (x,  y),  ...  converges  uniformly  in  R. 

*  Carathe"odory's  proof  is  given  in  a  paper  in  Schwarz- Festschrift,  and  in  Math.  Ann.  Bd.  LXXII, 
S.  107  (1912).,  Koebe's  proof  will  be  found  in  his  papers  in  Journ.  ftir  Math.  Bd.  CXLV,  S.  177 
(1915);  Acta  Math.  t.  XL,  p.  251  (1916). 

f  The  idea  of  equal  continuity  was  introduced  by  Aecoli,  Mem.  d.  R.  Ace.  dei  Lincei,  t.  xvm 
(1883). 


288  Conformed  Representation 

The  first  step  is  to  construct  a  sequence  of  points  Px,  P2,  ...  everywhere 
dense  in  R.  This  may  be  done  by  choosing  our  origin  outside  R  and  using 
for  the  co-ordinates  of  Ps  expressions  of  type 

XB  =  p2~",     y,=  p'2-"9     q>0 

where  p,  p'  and  q  are  integers,  and  where  the  index  s  =  /  (p,  p',  q)  is  a 
positive  integer  with  the  following  properties: 

1  (P>  Pr><l)>  I  (Po>  Po'>  9o)>  whenever  q  >  q0, 
I  (P>  />'»  ?)  >  I  (Po>  Po'>  7),  whenever  p  >  p0, 
L  (P,  P',  (l)  >  l  (P>  Po,  ?)»  whenever  p'  >  p0'. 

Since  the  functions  MS  (a;,  y)  are  bounded,  their  values  at  Pl  have  at  least 
one  limit  point  Ul  (xl,  y^.  We  therefore  choose  the  sequence  uln  (x,  y)  so 
that  it  converges  at  PL  to  this  limit  U1  (x1^  y^).  Since,  moreover,  the  func- 
tions uln  (x,  y)  are  uniformly  bounded  their  values  at  P2  have  at  least  one 
limit  point  U2(x2,y2)'.,  we  therefore  select  from  the  infinite  sequence 
u\n  (x>  y}  a  second  infinite  sequence  u2n  (x,  y)  which  converges  at  P2  to 
U2  (x<2,  y2)-  These  functions  u2n(x,y)  are  uniformly  bounded  and  their 
values  at  P3  have  at  least  one  limit  point  £/3  (x3, ?/3),  we  therefore  select 
from  the  sequence  u2n  (x,  y)  an  infinite  subsequence  u%n  (x,  y)  which  con- 
verges at  P3  to  C73  (x3,  ?/3),  and  so  on. 

We  now  consider  the  sequence  uu  (x,  ?/),  u22  (x,  y),  ^33  (^,  y), Since 

the  functions  are  all  equicontinuous  we  have 

I  ^m™  (*',</')  -  umm(x,y)  |  <  Je 
for  any  two  points  P  and  P'  whose  distance  PP'  is  not  greater  than  S. 

Next  let  2~q  be  less  than  8  and  let  r  be  such  that  the  set  P19  P2,  ...  Pr 
contains  all  the  points  of  R  for  which  q  has  a  selected  value  satisfying  this 
inequality,  then  a  number  N  can  be  chosen  such  that  for  m  >  N 

\Utt(Xs,ys)  ~  ttmmfon!/*)   I  <    !* 

for  all  values  of  Z  greater  than  m  and  for  all  points  (xs,  ys)  for  which  q  has 
the  selected  value.  This  number  ^V  should,  in  fact,  be  chosen  so  that  the 
sequences  umm  (x,  y)  converge  for  all  these  points  Ps.  These  points  P8 
form  a  portion  of  a  lattice  of  side  2~Q  and  so  there  is  at  least  one  of  these 
points  P'  within  a  distance  8  from  z.  We  thus  have  the  additional  in- 
equalities 

I  uu(x,y)    -  Uu   (x',yf)    |  <  K 

I  utt  (#',  y')  -  umm  (xf,  y')  |  <  $€, 
KU  (x,  y)    -  umm  (x,  y)    |  <  €. 

This  inequality  holds  for  all  points  P  in  R  and  proves  that  the  diagonal 

sequence  converges  uniformly  in  R  to  a  continuous  limit  function  u  (x,  y). 

There  is  a  similar  theorem  for  sequences  of  functions  of  any  number  of 

variables,  and  for  infinite  sets  of  functions  which  are  not  denumerable. 


Equicontinuity 

In  the  case  of  a  sequence  of  functions  /x  (2),  f2  (z),  ...  of  a  complex 
variable  z  =  x  -f  iy  there  is  equicontinuity  when 

I/.  (*')-/.  (3)  I  <  4* 

for  any  pair  of  values  z,  z'  for  which  |  z'  —  z  \  <  S,  S,  as  before,  being 
independent  of  s  and  of  the  position  of  z  in  the  region  E.  A  sufficient 
condition  for  equicontinuity,  due  to  Arzela*,  is  that 


Z'-Z  < 


for  all  functions  fs  (z)  of  the  set  and  for  all  pairs  of  points  z,  z'  of  the  domain, 
Ml  being  a  number  independent  of  s,  z  and  z' '.  We  need  in  fact  only  take 
SJ^j  =  ^e  to  obtain  the  desired  inequality. 

In  the  particular  case  when  each  function  /s  (z)  possesses  a  derivative 
it  is  sufficient  for  equicontinuity  that  |//  (z)  |  <  M2,  where  M2  is  inde- 
pendent of  s  and  z.  The  result  follows  from  the  formula  for  the  remainder 
in  Taylor's  theorem. 

Montel|  has  shown  that  if  a  family  of  functions  fs  (z)  is  uniformly 
bounded  in  a  region  R  it  is  equicontinuous  in  any  region  R'  interior  to  R. 

Suppose,  in  fact,  that  |  /,  (z)  |  <  M  for  any  point  z  in  R  and  for  any 
f  unction  fs  (z)  of  the  family,  the  suffix  s  being  used  simply  a.s  a  distinguishing 
mark  and  not  as  a  representative  integer. 

Let  D  be  a  domain  bounded  by  a  simple  rectifiable  curve  G  and  such 
that  R  contains  D  while  D  contains  R'.  We  then  have  for  any  point  £ 
within  R' 

,,  ,w         If  f* (z)dz 


Therefor* 


where  I  is  the  length  of  C  and  h  is  the  lower  bound  of  the  distance  between 
a  point  of  G  and  a  point  of  R'  .  This  inequality  shows  that  the  functions 
fs  (z)  are  equicontinuous  in  R'  '. 

Now  if  f8  (z)  =  us  (x,  y)  -f-  iv8  (x,  y)  where  ua  and  vs  are  real,  the  func- 
tions us,  vs  are  likewise  equicontinuous  and  uniformly  bounded  in  R'  .  We 
can  then  select  from  the  set  ua  a  sequence  uu  ,  w22  ,  u^  ,  .  .  .  which  converges 
uniformly  to  a  function  u  (x,  y)  which  is  continuous  in  R'.  Also  from  the 
associated  sequence  vn  ,  v22  ,  v&  ,  .  .  .  we  can  select  an  infinite  subsequence 
vaa,  vbb,  vce,  ...  which  converges  uniformly  in  R'  to  a  continuous  function 
v  (x,  y).  The  series 

fa*  (*)  +  [/*>  (Z)  -  faa  (*)]  +  L/cc  (*)  ~  /»  (*)]  +   ... 

then  converges  uniformly  to  u  (x,  y)  +  fy  (a;,  y),  which  we  shall  denote  by 

*  Mem.  delta  R.  Ace.  di  Bologna  (5),  t.  vm. 
t  Annales  de  Vtfcole  Normale  (3),  t.  xxiv,  p.  233  (1907). 
B  19 


290  Conformal  Representation 

the  symbol  /  (z).  On  account  of  the  uniform  convergence  the  function 
/  (z)  is,  by  Weierstrass'  theorem,  an  analytic  function  of  z  in  J?'.  Indeed  if 
/,(n)  (z)  denotes  the  nth  derivative  of  any  function  f8  (z)  of  our  set  and  C' 
is  any  rectifiable  simple  closed  curve  contained  within  J?',  we  have  by 
Cauchy's  theorem  and  the  property  of  uniform  convergence 

faa(n)(z)  +  [fn(n)(z)~faa(n)  (*)]+••• 

n\ 


Since/  (£)  is  continuous  in  J?'  and  on  C'  the  integral  on  the  right  represents 
an  analytic  function.  When'n  =  0  this  tells  us  that  the  sequence 

/oa  («)>/»  (*),  — 

converges  to  an  analytic  function  which,  of  course,  is  /  (z).  When  n  ^  0  the 
relation  tells  us  that  the  sequence  /aa(n)  (z),  fbb(n)  (z),  ...  converges  to/(n)  (z). 
We  may  conclude  from  Cauchy's  expression  for  fs(n)  (z)  as  a  contour 
integral  that/s(n)  (z)  is  uniformly  bounded  in  any  region  containing  Rr  and 
contained  in  R.  It  then  follows  that  the  set/s(n)  (z)  is  equicontinuous  in  R  . 
Hence  from  the  sequence  faa  (z),  fbb  (z),fcc  (z),  ...  we  can  select  an  infinite 
sequence  faa  (z),  fftft  (z),/yy  (z),  ...  such  that  /««'  (z),fftftr  (z),/y/  (z),  ...  con- 
verges uniformly  in  R'  to  an  analytic  function  which  can  be  no  other 
than  /'  (z).  At  the  same  time  the  sequence  faa  (z),/^  (z),  ...  converges  uni- 
formly to/(z).  This  process  may  be  repeated  any  number  of  times  so  as 
to  give  a  partial  sequence  of  functions  converging  uniformly  to  /  (z)  and 
having  the  property  that  the  associated  sequences  of  derivatives  up  to  an 
assigned  order  n  converge  uniformly  to  the  corresponding  derivatives  of 


We  now  consider  a  sequence  of  contours  Cl)  (72,  ...  Cn  having  for 
limit  the  contour  CQ  which  bounds  R,  the  contours  Cl9  (72,  ...  Cn  bounding 
domains  Dl9D%9  ...,  each  of  which  contains  the  preceding  and  has  .R  as 
limit.  From  our  set  of  f  unctions  fs  (z)  we  can  select  a  sequence  fsl  (z)  which 
converges  uniformly  in  Dl  towards  a  limit  function,  from*  the  sequence 
/«i  (z)  we  can  cun*  a  new  sequence  f82  (z)  which  converges  uniformly  in  D2 
to  a  limit  function  and  so  on.  The  diagonal  sequence  /n  (z),/22  (z),  ...  then 
converges  uniformly  throughout  the  open  region  .R  to  a  limit  function. 

Hence  we  have  Montel's  theorem  that  an  infinite  set  of  uniformly 
bounded  analytic  functions  admits  at  least  one  continuous  limit  function, 
both  boundedness  and  continuity  being  understood  to  refer  to  the  open 
region  R  in  which  the  functions  are  defined  to  be  analytic. 

For  further  developments  relating  to  this  important  theorem  reference 
must  be  made  to  Montel's  paper.  For  the  case  of  functions  of  a  real  variable 


Mapping  of  an  Open  Region  291 

A.  Roussel*  has  recently  invented  a  new  method.  The  selection  theorem 
has  been  extended  by  Montel  to  functions  of  bounded  variation. 

§  4-46.  Mapping  of  an  open  region.  Let  B  be  a  simply  connected  bounded 
region  which  contains  the  origin  0  and  has  at  least  two  boundary  points. 
Let  S  be  the  set  of  analytic  functions  fs  (z)  which  are  uniform,  regular, 
smooth  and  bounded  in  R  and  for  which 

/s(0)  =  o,  /;(0)=i,    \f.(z)\<M. 

Let  Us  be  the  upper  limit  of  |  fs  (z)  \  in  R  and  let  p  be  the  lower  limit  of  all 
the  quantities  Us.  There  is  then  a  sequence  fr  (z)  of  the  functions/,  (z)  for 
which  Ur  ->  p.  Since,  moreover,  this  sequence  is  uniformly  bounded,  we 
can  apply  the  selection  theorem  and  construct  an  infinite  subsequence 
which  converges  uniformly  to  a  limit  function  /  (z)  in  any  closed  partial 
region  R'  or  R.  This  function  /  (z)  is  a  regular  analytic  function  in  R  and 
satisfies  the  conditions/  (0)  =  O,/'  (0)  =  1.  Being  a  uniform  limit  function 
of  a  sequence  of  smooth  mapping  functions  it  is  smooth  in  R  and  its  U  is  p. 

The  function  /  (z)  thus  maps  R  on  a  region  T  which  lies  in  the  circle 
with  centre  O  and  radius  p.  If  T  does  not  completely  fill  the  circle,  there 
will  be  a  value  re**,  with  r  <  p,  which  is  not  assumed  by  our  function  /  (z) 
in  R.  We  shall  now  show  that  this  is  impossible  and  that  consequently  T 
does  fill  the  circle. 

Let  r  =  a2p,  then  a  <  1  and  if  we  write 

f  (z]  -      2a     0(>i*  J7  (*)  -JL 
h(Z}~  l  +  a*pe    av(z)~-  1' 

r    /  v-,0     /(z)  -  tfpe* 

where  [f  (*  ]   --zfT  N     ~    ^  > 

L    v  /J        a2/  (z)  —  pe*v 

v  (0)  =  a, 

we  have  /0  (0)  =  0,  /</  (0)  =  1  and  the  function  /0  (z)  is  uniform,  regular, 
smooth  and  bounded  in  R. 

Now  let  UQ  be  the  upper  limit  of /0  (z)  in  R.  We  may  find  an  inequality 
satisfied  by  this  quantity  by  observing  that 

v  (z)  —  a 
av  (z)  —  1 

is  of  the  form  ri/r2,  where  rly  r2  are  the  distances  of  the  point  v  (z)  from  the 
points  a,  I/a  respectively  which  are  inverse  points  with  respect  to  the 
circle  |  z  \  =  1. 

On  the  other  hand, 

a2  |  v(z)  |2  =  P!//^, 

where  pl9  p2  are  the  distances  of  the  point  f  (z)  from  the  points  a^pe**, 
a"2pe^  which  are  inverse  points  with  respect  to  the  circle  |  z  \  —  p.  Now 

*  Journ.  de  Math.  (9),  t.  v,  p.  395  (1926).  See  also  Bull,  des  Sciences  Math.  t.  LKL,  p.  232  (1928). 

19-2 


292  Conformal  Representation 

the  point  /  (z)  lies  either  on  this  circle  or  within  it  and  so  p!/p2  has  a  value 
which  is  constant  either  on  |  z  \  =  p  or  on  a  circle  within  |  z  \  =  p  and  with 
the  same  pair  of  inverse  points.  This  constant  for  a  circle  with  this  pair  of 
inverse  points  has  its  greatest  value  for  the  circle  |  z  \  —  p  if  circles  lying 
outside  this  circle  are  excluded.  This  greatest  value  is,  moreover, 

^^  -  ^ 

p  -  a~2p 

Hence  we  have  the  inequality  |  v  (z)  |2<  l.  By  a  similar  argument  we 
conclude  that  rx/r2  has  its  greatest  value  when  the  point  v  (z)  is  at  some 
place  on  the  circle  |  z  \  =  1  and  this  value  is  a.  Hence 

v  (z)  —  a 
av  (z)  -  1 

and  so  U(}  <  p.  We  have  thus  found  a  function  for  which  C70  <  p,  and  this 
is  incompatible  with  the  definition  of  p  as  the  lower  limit  of  the  quantities 
U K.  The  region  T  must  then  completely  fill  the  circle  of  radius  p  and  so  the 
function/  (z)  maps  E  on  this  circle.  The  radius  p  is  consequently  called  the 
radius  of  the  region  R. 

This  analysis,  which  is  due  to  L.  Fejcr  and  P.  Riesz,  is  taken  from  a 
paper  by  T.  Rado*.  The  analysis  has  been  carried  further  by  G.  Julia|  who 
first  selects  from  the  functions  fs  (z)  the  polynomials  p^(n)  (z)  of  degree  n. 
Among  these  polynomials  there  is  one  polynomial  p(n}  (z)  whose  maximum 
modulus  has  a  minimum  value  mn.  It  is  clear  that  mn  >  p.  Julia  specifies 
a  type  of  region  R  for  which  the  sequence  p(n)  (z)  possesses  a  limit  function 
/  (z)  mapping  the  region  R  on  the  circle  of  radius  p. 

§  4-51.  Conformal  representation  and  the  Green's  function.  Consider  in 
the  .T?/-plane  a  region  A  which  is  simply  connected  and  which  contains  the 
origin  of  co-ordinates.  We  shall  assume  that  the  boundary  of  A  is  smooth 
bit  by  bit.  We  write 


for  the  Green's  function  associated  with  the  origin  as  view-point,  r  being 
short  for  (a:2  -f  ?/2)4.  Let  us  write  H  (0,  0)  =  log  r  log  (!//>),  then  r  is  the 
capacity  constant  or  constant  of  Robin J.  Now  let 

Z=  <f>(z)  =  z  -f  c2z  +  c3z3  -|-  ... 

be  the  uniquely  determined  function  which  maps  the  interior  of  a  circle 
|  Z  |  <  p  on  A  in  such  a  manner  that 

<f>  (0)  =  0,     c£'  (0)  =  1. 

*  Szeged  Ada,  1. 1,  p.  240  (1923). 
|  Complex  Rendus,  t.  CLXXXIIT,  p.  10  (1926). 

J  The  boundafy  of  A  may  also  be  taken  to  be  a  closed  Jordan  curve,  in  which  case  r  is  the 
transfmite  diameter. 


Relation  to  the  Green's  Function  293 

It  will  be  shown  that  the  Green's  function  G  (x,  y)  is 
0  (x,  y)  =  log  (p/r),     f=|^(2)|. 

Bieberbach*  has  proved  a  theorem  relating  to  the  area  of  the  region  A 
which  is  expressed  by  the  inequality  area  >  Tip2.  This  means  that  among 
all  regions  A  for  which  H  (0,  0)  has  a  prescribed  value  the  circle  possesses 
the  smallest  area. 

For  the  theorem  relating  to  the  Green's  function  we  may,  with  ad- 
vantage, adopt  a  more  general  standpoint.  Let  us  suppose  that  the 
transformation  w  =  /  (z)  maps  the  area  A  on  the  interior  of  a  unit  circle 
in  the  w-plane  in  such  a  way  that  to  each  point  of  the  circle  there  corre- 
sponds only  one  point  of  the  area  A  and  vice  versa.  Let  the  centre  of  the 
circle  correspond  to  the  point  z0  of  the  area  A ,  then  z0  is  a  simple  root  of 
the  equation /  (z0)  ==  0  and/  (z)  =  0  has  no  other  root  in  the  interior  of  A. 
This  is  true  also  for  the  boundary  if  it  is  known  that  there  is  a  (I,  1) 
correspondence  between  the  points  of  the  unit  circle  and  the  points  of  the 
boundary  of  A .  We  may  therefore  write 

'fW-k-zJe*™, 

where  the  function  p  (z)  is  analytic  in  A. 

Putting  p  (z)  =  P  +  iQ,  z  —  ZQ  =  re1*,  where  P,  Q,  r  and  6  are  all  real, 
we  have 

w=f(z)  =  exp  {log  r  +  P  +  i  (Q  +  9)}. 

Now,  by  hypothesis,  the  boundary  of  A  maps  into  the  boundary  of  the 
unit  circle,  therefore  log  r  +  P  must  be  zero  on  the  boundary  of  A .  This 
means  that  log  r  -f-  P  is  a  potential  function  which  is  infinite  like  log  r  at 
the  point  (x0,  ?/()),  is  zero  on  the  boimdary  of  A  and  is  regular  inside  A 
except  at  (#0,  yQ).  This  potential  has  just  the  properties  of  the  function 
G  (x,  y\  XQ,  3/0),  where  G  (x,  y,  x0,  y0)  is  the  Green's  function  for  the  area  A 
when  (x0,  y0)  is  taken  as  view-point,  consequently  the  problem  of  the 
conformal  mapping  of  A  on  the  unit  circle  is  closely  related  to  that  of 
finding  the  function  G. 

.Writing  a  =  Q  -f-  0,  0  <  a  <  2-rr,  we  have  on  the  boundary  of  the  circle 

dw  =  iel*dcr, 

while  on  the  boundary  of  A 

dz=  \dz\  e*+, 

where  ifj  is  the  angle  which  the  tangent  makes  with  the  real  axis.    Since 
dzfdw  is  neither  zero  nor  infinite,  the  function 

dz\ 
dw) 


—  i  log  ( i 


L.  Bieberbach,  Rend.  Palermo,  vol.  xxxvm,  p.  98  (1914).  This  theorem  is  discussed  in  §  4-91. 


294  Conformal  Representation 

is  analytic  within  the  circle  and  its  real  part  takes  the  value  ^  —  a  on  the 
boundary  of  the  circle.   On  the  other  hand,  the  function 

F  (w)  =  -  i  log  [-  i  (1  -  w)2  dz/dw] 

is  analytic  within  the  circle  and  its  real  part  takes  the  value  iff  on  the 
boundary. 

If  0  is  a  known  function  of  a  on  the  boundary  of  A,  Schwarz's  formula 
gives  , 

0 


where  k  is  an  arbitrary  constant.  The  preceding  formula  then  gives  z  by 
means  of  the  equation  ^  M  ^ 

_     <•?       —     n      I  _     _ 

(O       -      V      I  -j~  ~\~f)  • 

)W>(1  -  w)* 

The  relation  between  </r  and  a  is  partly  known  \fhen  the  boundary  of 
A  is  made  up  of  segments  of  straight  lines  but  in  the  general  case  i/j  is  an 
unknown  function  of  o-  and  the  present  analysis  gives  only  a  functional 
equation  for  the  determination  of  0. 

To  see  this  we  suppose  that  on  the  circumference  of  the  circle 

F  =  i/j  4-  ;</> 

where  <f>  and  i/j  are  real,  then 

|  dz  |  =  -J  cosec2  -  |  da  |  e~^, 

2i 

and  the  curvature  of  the  boundary  of  A  is 


and  may  be  regarded  as  a  known  function  of  j/r,  say  (7  (i/r).   Making  use  of 
the  relation  between  cf>  and  i/j  of  §  3'  33 

*  (a)  =  ^  ~  27T  r  ^'  (a<>)  iog  [^  c°sec2  a°  ?~]  da°' 

where  6  is  a  constant,  we  obtain  the  functional  equation* 


4(7(0)  sin^ 
where  0  (a)  is  defined  by  the  foregoing  equation. 

EXAMPLE 
Prove  that 


[K.  Lowner.] 

§4-61.  Scliwarz's  lemma.  It  was  remarked  that  a  Taylor  expansion  for 
/  (w)  in  powers  of  w  —  w0  is  not  required  for  points  w0  on  the  real  axis, 

*  T.  Levi  Civita,  Rend.  Palermo,  vol.  xxni,  p.  33  (1907);  H.  Villat,  Annaies  de  rtfcole  Normale, 
t.  xxvm,  p.  284  (1911);  U.  Cisotti,  Idromeccamca  piana,  Milan,  p.  50  (1921). 


Schwarz^s  Lemma  295 

but  when  f(w)  is  real*  for  real  values  of  w  belonging  to  a  finite  interval, 
Schwarz  has  shown  that  it  is  possible  to  make  an  analytical  continuation 
of  /  (w)  into  a  region  for  which  v  is  negative.  Let  us  consider  an  area  S 
bounded  by  a  curve  ACB  of  which  the  portion  A  B  is  on  the  line  v  =  0 
within  the  interval  just  mentioned. 

Let  S'  be  the  image  of  S  in  the  line  v  =  0  and  let  the  value  otf(w)  for 
a  point  w'  of  S'  be  defined  as  follows.   We  write 

w  =  u  +  iv9     w'  —  u  —  iv, 


where  u,  v,  f,  77  are  all  real.  The  function  f  (w)  being  now  defined  within 
the  region  S  +  S'  we  write 

g  (w,  £)  =  1/27T*  (w  -  0 
and  consider  the  two  integrals 


=       9 


f  =  f    (7  (">, 

Js' 


taken  round  the  boundaries  of  S  and  S'     Since  /  (w)  is  analytic  within 
both  S  and  $'  we  have 

/=/(£)>     /'  =  0       when  £  lies  within  S, 
7=0,          /'=/(£)  when  £  lies  within  S'. 
Hence  in  either  case  7  +  /'=/(£)  and  so 

/(£)=(        <7(t0,£)/(MOdt0, 


for  the  two  integrals  along  the  line  ^4  J3  are  taken  in  opposite  directions  and 
so  cancel  each  other. 

Now  the  integral  in  this  equation  can  be  expanded  in  a  Taylor  series 
of  ascending  powers  of  £  -  £0  for  any  point  £0  within  the  area  S  -f  S' 
whether  £0  is  on  the  real  axis  or  not.  The  integral  in  fact  represents  a 
function  which  is  analytic  within  the  area  S  f  S'  and  can  be  used  to  define 
/(£)  within  S  +  /S".  In.  this  case,  when  £0  is  on  AB,  f  (£)  can  be  expanded 
in  a  power  series  of  the  foregoing  type  and  the  coefficients  in  this  series, 
being  of  type 


are  all  real. 

•  *  It  is  assumed  here  that  /  (w)  has  a  definite  finite  real  integrable  value  for  these  real  values 
of  w.  In  a  recent  paper,  Bull,  des  Sciences  Math.  t.  LIT,  p.  289  (1928),  G.  Valiron  has  given  an 
extension  of  Schwarz's  lemma  in  which  it  is  simply  assumed  that  the  imaginary  part  irj  of  f  (w) 
tends  uniformly  to  zero  as  v  ->  0.  If,  then,  the  function  /  (w)  is  holomorphic  in  the  semicircle 
|  w  \  <  R,  v  >  0,  it  is  holomorphic  in  the  whole  of  the  circle  |  w  \  <  R. 


296  Conformal  Representation 

Let  us  now  use  z0  to  denote  the  value  of  z  corresponding  to  this  value 
£0  of  w.  The  equation 

z  -  z,  +-f(w)  -/(U  =  a  (w  -  £0)  +  b  (w  -  £0)*  +  c  (w  -  &)»  4-  ... 
can  be  solved  for  w  —  £0  by  the  reversion  of  series  if  a  ^  0,  and  the  series 
thus  obtained  is  of  type 

w-£0  =  A(z-z0)  +  B(z-  2o)2  +  C  (z  -  z,,)3  +  ...  , 

where  the  coefficients  A,  B,  C  are  all  real.  The  exceptional  case  a  —  0 
occurs  only  when  the  correspondence  between  w  and  z  at  the  point  £0 
ceases  to  be  uniform. 

§  4-62.  The  mapping  function  for  a  polygon.  Let  us  now  consider  an 
area  A  in  the  z-plane  which  is  bounded  by  a  contour  formed  of  straight 
portions  L19  L2,  ...  Ln.  Let  z0  denote  a  point  on  one  of  the  lines  L  and 
let  tin  be  the  angle  which  this  line  makes  with  the  real  axis,  also  let  WQ  be 
the  value  of  w  corresponding  to  z. 

It  is  easily  seen  that  the  function 

/  (w)  -  (z  -  z0)  e-*- 

has  the  properties  of  a  mapping  function  for  points  z  within  A,  and 
consequently  also  for  the  corresponding  region  in  the  w-plane;  it  is  real 
when  the  point  z  is  on  the  line  L  in  the  neighbourhood  of  z0  and  changes 
sign  as  z  passes  through  the  value  z0;  consequently,  when  considered  as 
a  function  of  w  it  is  real  on  the  real  axis  and  changes  sign  as  w  passes 
through  the  value  w0.  Schwarz's  lemma  may,  then,  be  applied  to  this 
function  to  define  its  continuation  across  the  real  axis  and  it  is  thus  seen 
that  we  may  write 

e-ih«  (z  _  2())  =(20-  WQ)  P  (W  -  W0), 

where  P  (w  —  WQ)  denotes  a  power  series  of  positive  integral  powers  of 
w  —  WQ  including  a  constant  term  which  is  not  zero.  From  this  equation 
it  follows  that  in  the  neighbourhood  of  the  point  w0 


where  P0  (w  —  w0)  is  real  when  w  and  w0  are  real. 

Taking  logarithms  and  differentiating  again,  we  see  that  the  function 


=   *  (log  &\ 
dw\    *  dw) 


is  real  and  finite  in  the  neighbourhood  of  w  =  WQ. 

Next,  let  zl  denote  the  point  of  intersection  of  two  consecutive  lines 
L,  L' ',  intersecting  at  an  angle  0:77;  the  argument  of  zl  —  z  varies  from  hrr 
to  fnr  —  «TT  as  the  point  z  passes  from  the  line  L  to  L'  through  the  point  of 
intersection  (Fig.  19).  Hence  the  function 

1 
J=  [(«!  -  z)e~lh*]a 


Mapping  Function  for  a  Polygon  297 

is  real  and  positive  on  L  and  negative  on  L' .  Moreover,  it  has  the  required 
properties  within  A ,  and  when  considered  as  a       _, 
function  of  w  it  has  the  required  mapping  pro- 
perties in  the  region  corresponding  to  A  and 
is  real  on  the  real  axis.  By  Schwarz's  lemma 
we  may  continue  this  function  across  the 
real  axis  and  may  write  for  points  w  in  the 
neighbourhood  of  w1 , 

J  =  (w  -  Wi)  PI  (w  -  Wi), 

where  P1  (w  —  w±)  is  a  power  series  with  real 
coefficients  and  with  a  constant  term  which  is 
not  zero.  This  equation  gives 

z  —  zl  =  e*hir  (w  —  Wi)a  P2  (w  —  w^, 

where  P2  (w  —  w±)  is  another  power  series  with  real  coefficients.  This 
equation  indicates  that  for  points  in  the  neighbourhood  of  wl 

dZ  ..         .  x         1     T>       /  x 

^  =  «*•<«- «*>•-*  pa(»-«a 

where  P3  (w  —  w±)  is  a  power  series  with  real  coefficients.  Taking  logarithms 
and  differentiating  we  find  that 

n  /    x        d   /,       dz\        a  —  1        m  ,  . 

F  (w)  =  j-  ( log  j  - )  =     h  T  (w  ~  WJ, 

dw  \    °  dw)      w  —  w^ 

where  T  (w  —  wj  is  a  power  series  with  real  coefficients.  The  function 

F  (      -    a~  l 
w  —  Wi 

is  thus  analytic  in  the  neighbourhood  of  w  —  wl . 

For  a  point  z2  on  the  boundary  of  A  which  corresponds  to  ^v  we  have 
(if  z2  is  not  a  corner  of  the  polygon) 

2       w       w2 

Therefore  -T-  = lp\~ 

dw          w2  ^  \w. 


d   A       dz\  2        1        /IN 

=  j-  (  log  3-  )  =  ----  h  —  s  P!  I  -   , 
d?/;  \    &  dw;/  te;      w;2  /    \w;/ 


where  p  (l/w)  is  a  power  series.  The  expansion  for  z  —  z2  may,  indeed,  be 
obtained  by  mapping  the  half  plane  w  into  itself  by  means  of  the  sub- 
stitution w  —  —  \IW-L  ,  and  by  then  using  the  result  already  obtained  for 
an  ordinary  point  z0  on  L. 

The  function  F  (w)  is  real  for  all  real  values  of  w,  as  the  foregoing 
investigation  shows,  is  analytic  in  the  whole  of  the  upper  half  of  the 
w-plane  and  is  real  on  the  real  axis,  the  fact  that  it  is  analytic  being  a 


298  Conformal  Representation 

consequence  of  the  supposition  that  the  inverse  function  z  —  g  (w)  is 
analytic  in  the  upper  half  of  the  w-plane.  Applying  Schwarz's  lemma  we 
may  continue  this  function  F  (w)  across  the  real  axis  and  define  it  analytically 
within  the  whole  of  the  w-plane,  the  points  on  the  real  axis  which  are 
poles  of  F  (w)  being  excluded. 

When  |  w  \  is  large  |  F  (w)  \  is  negligibly  small,  as  is  seen  from  the 
expansion  in  powers  of  ljw\  moreover,  F  (w)  has  only  simple  poles  corre- 
sponding to  the  vertices  of  the  polygon  A  and  these  are  finite  in  number. 
Hence,  since  F  (w)  outside  these  poles  is  a  uniform  analytic  function  for 
the  whole  w-plane,  it  must  be  a  rational  function. 

Let  a,  6,  c,  ...  1  be  the  values  of  w  corresponding  to  the  vertices  of  the 
polygon  and  let  arr,  /?TT,  ...  XTT  be  the  interior  angles  at  these  vertices,  then 

™  /    x       ^  a  —  I        d  ,  ,     dz 

F  (w)  =  2 =  -=-  log  -,—  , 

w  —  a      aw     &  aw 

and  there  is  a  condition 

S  (a  -  1)  =  -  2, 

which  must  be  introduced  because  t}ie  sum  of  the  interior  angles  of  a 
closed  polygon  with  n  vertices  is  equal  to  (n  —  2)  TT. 
Integrating  the  differential  equation  for  z  we  obtain 

z  =  c(  (w  -  a)"-1  (w  -  &V3-1 ...  (w  -  I)*"1  dw  +  C", 

where  C  and  C"  are  Arbitrary  constants.  By  displacing  the  area  A  without 
changing  its  form  or  size  but  perhaps  changing  its  orientation  we  can 
reduce  the  equation  to  the  form 


=  K 


\(w-  a)*-1  (w  -  by-*  ...(w-  I}*-1  dw, 


where  K  is  a  constant.  This  is  the  celebrated  formula  of  Schwarz  and 
Christoffel*  If  one  of  the  angular  points  with  interior  angle  fin  corresponds 
to  an  infinite  value  of  w,  the  number  of  factors  in  the  integrand  is  n  —  1 
instead  of  n,  and  the  equation 

S  (a  -  1)  =  -  2 
may  be  written  in  the  form 

S  (a  -  1)  =  -  1  -  p, 

where  now  the  summation  extends  to  the  n  —  1  values  of  a  which  appear 
in  the  integral. 

Since  we  can  choose  arbitrarily  the  values  of  w  corresponding  to  three 
vertices  of  the  polygon,  there  are  still  n  —  3  constants  besides  C  and  C' 
to  be  determined  when  the  polygon  is  given.  In  the  case  of  the  triangle 
there  is  no  difficulty.  We  can  choose  a,  6  and  c  arbitrarily;  a,  ]8  and  y  are 
known  from  the  angles  of  the  triangle  and  by  varying  K  we  can  change 
the  size  of  the  triangle  until  the  desired  size  is  obtained. 

*  E.  B.  Christoffel,  Annali  di  Mat.  (2),  1. 1,  p.  95  (1867);  t.  iv,  p.  1  (1871);  Get.  Werke,  Bd.  I, 
S.  265.  H.  A.  Schwarz,  Journ.filr  Math.  Bd.  LXX,  S.  105  (1869);  Ges.  Abh.  Bd.  n,  S.  65. 


Mapping  of  a  Triangle  299 

An  interesting  example  of  the  conformal  representation  of  a  triangle 
with  one  corner  at  infinity  is  furnished  by  the  equation 

z  —  z0  =  \    f  (>s)  ds,  where  /  (6*)  =  (2a/7r)  (1  —  s2)%/s,  w  =  u  ~\-  iv. 
When  w  lies  between  1  and  oo  we  have 


=  ft  -f  ic,  say, 

where  ft  is  a  constant  and  c  varies  from  0  to  oo.  Thus,  the  portion  w  >  1 
of  the  real  axis  corresponds  to  a  line  parallel  to  the  axis  of  y. 
Again,  if  0  <  w  <  1,  we  may  write 


i 

-  ft  -  d, 

where  d  varies  from  0  to  06.  The  portion  0  <  w  <  1  of  the  real  axis  corre- 
sponds, then,  to  a  line  parallel  to  the  axis  of  x  and  extending  from 

z  --=  z(i  -f-  ft  to  —  oo. 

When  —  1  <  w  <  0  we  may  write 

r  -  1  no 

*-*o=         f(*)ds+         f(s)ds 
Ji  J  -i 

=  ft'  4-  d', 

and  so  the  corresponding  line  in  the  z-plane  extends  from  —  oo  to  ft'  H-  z0 
and  is  parallel  to  the  axis  of  x.    When  —  oo  <  w  <  1,  we  have 


f  ~l  (~l 

-  ZD  =         /  (*)  ds-  \      f  (s)  ds 

Ji  Jw 


-  ft'  -  ic', 

where  c'  ranges  from  0  to  oo,  and  so  this  part  of  the  ^-axis  corresponds 
to  a  line  from  ft'  parallel  to  the  i/-axis.  The  two  lines  parallel  to  the  i/-axis 
can  be  shown  to  be  portions  of  the  same  line  separated  by  a  gap.  We  have 
in  fact 

b-bf=ff  (s)  ds  -  f  "V  (s)  ds  =  f1    /  (s)  ds, 

Ji  Ji  J  -I 

where  the  integral  is  taken  along  the  semicircle  with  the  points  —  1,  +  1 
as  extremities  of  a  diameter.  On  this  semicircle  we  may  put 


andso 


f°  i 

(7r/2a)  (ft  -  ft')  =  i\    -*•  d0  [-  2i  sin  0  e^ 

J  re 

fir   /  /3  /D\ 

-=  i  (1  -  i)      (cos  -  +  i  sin  -  J  (sin  6)4  rf0 

-  2i  f"  cos  ?  (sin  9)%  d0  =  2i^2T  (J)  T  (f  )  =  i. 


300  Conformal  Representation 

The  figure  in  the  z-plane  is  thus  of  the  type  shown  in  Fig.  20.  To  solve 
an  electrical  problem  with  the  aid  of  this  transformation  we  put  d>  =  ie^x, 
where  %  is  the  complex  potential  $  -f  ifi.  This  transformation  maps  the 
half  w-plane  for  which  v  >  0  on  a  strip  of  the  ^-plane  lying  between  the 
lines  <f>  —  il:  77. 

Performing  the  integration  we  find  that 


log  ?-±-     -  2  V2  -  2  log 


where 


Fig.  20. 


-C, 


1-1 


Fig.  21. 


When  the  real  part  of  w  is  large  and  negative  the  chief  part  of  the 
expression  for  z  is 


a 


log  (r  -  1)  =  -      log  (1  +  &  +  ...  -  1)  =      log  2  -  - 


77 


77 


This  gives  a  field  that  is  approximately  uniform.  On  the  other  hand, 
when  the  real  part  of  w  is  large  and  positive,  the  chief  part  of  the  expression 
for  z  is 


and  we  may  thus  get  an  idea  of  the  nature  of  the  field  at  a  point  outside 
the  gap  and  at  some  distance  from  its  surfaces.  These  results  are  of  some 
interest  in  the  theory  of  the  dynamo.  Another  interesting  example,  in 
which  the  polygon  is  originally  of  the  form  shown  in  Fig.  21,  gives  edge 
corrections  for  condensers*. 

Assigning  values  of  w  to  the  corners  in  the  manner  indicated,  the 
transformation  is  of  type 

\   1  [(w  +   ^  (w  -  b)  dw 
CaC^Ce)-*  V  ----  -'-+--         ' 
J(w  —  0,1)*  (w  - 


n 

z  -  G 


--  ----  — 

(w  -f  a2)  * 


r.  . 

[(q  -  w)  ...  (c6  + 


*  J.  J.  Thomson,  Recent  Researches  in  Electricity  and  Magnetism,  1893;  Maxwell,  Electricity 
and  Magnetism,  French  translation  by  Potier,  ch.  n,  Appendix;  J.  G.  Coffin,  Proc.  Amer.  Acad. 
of  Arts  and  Sciences,  vol.  xxxix,  p.  415  (1903). 


Edge  Correction  for  a  Condenser  301 

Making  a8  ->  0,  c8  ->  oo  we  finally  obtain 


where  G  and  6  are  constants  to  be  determined. 
Integrating,  we  find  that 

z  =  C  [w  +  I  (w  -  6)2  -  6  log  (-  w)  +  F], 

where  F  is  a  constant  of  integration.   Since  z  =  0  when  w?  =  —  1,  we  have 

F=  1-  |(1  +  6)2. 

When  ^  is  small  and  positive,  the  imaginary  part  of  z  must  be  ^7?,  and 
the  real  part  must  be  negative.  Since  the  argument  of  —  w  in  both  con- 
ditions are  satisfied  by  taking  C  =  —  h/bir,  therefore 

_         h 

Assuming  that  the  potential  </>  is  zero  on  A  A  and  equal  to  V  on  BB1 , 
we  may  write 

^  =  0  —  i<^  =        (log  WJ  —  ITT). 

7T 


Fig.  22. 

The  charge  per  unit  length  on  BB'  from  the  edge  (w  =  6)  to  a  point 
P  (?#  =  s)  so  far  from  B  that  the  surface  density  is  uniform  is 

1  V 

q==  ~~        ^p  ~  ^  =  ~         log  (<S/6)l 


47T 
Now  when  5  is  very  small  and  positive,  z  =  x  -f  i&,  and  so 

x  +  iA  =  _  ---  (i  -  26  -  2i&7r  -  26  log  5). 


Therefore  log  (s/b)  =  irx/h  +  1/26  -  1  -  log  6, 

F     TTX      1  -  26 


, 
and  so  8s= 

When  b  =  1  we  have  the  well-known  result 


,      /] 
_log6j. 


in  which  it  must  be  remembered  that  x  is  negative. 


302  Conformal  Representation 

When  w  is  very  small  and  negative,  z  —  x,  and  so 


2  Io 

V    P 

W>,  LTT 

V      h 


,rl        ,        ,                        V    (h          \ 
VV  hen  b  —  \         q  —  —   .  -.  I x  } , 

47T//,    \277  / 

A  f/      (h  \ 

6-00       r/  =  -  —  -       -  x }. 

A~L    \7T  ] 


Since  ?^  --  6  at  the  point  B,  the  value  of  2  for  this  point  is 
3  =  -  5f-  [1  -  62  -  26  log  6  -  26i7r], 

2u7T 

and  so  the  upper  plate  projects  a  distance  d  beyond  the  lower  one,  where 


Many  important  electrostatic  problems  relating  to  condensers  are  solved 
by  means  of  conforinal  representation  in  an  admirable  paper  by  A.  E.  H. 
Love*.  The  problem  of  the  parallel  plate  condenser  is  treated  for  planes 
of  unequal  breadth  and  for  planes  of  equal  breadth  arranged  asymmetrically. 
The  formulae  involve  elliptic  functions.  The  hydrodynamical  problems 
relating  to  two  parallel  planes,  when  the  motion  is  discontinuous,  are 
treated  in  a  paper  by  E.  G.  C.  Poolef. 

Some  applications  of  conformal  representation  to  problems  relating  to 
gratings  are  given  in  a  paper  by  H.  W.  Richmond^.  The  general  problem 
of  the  conformal  mapping  of  a  plane  with  two  rectilinear  or  two  circular 
slits  has  been  discussed  recently  by  J.  Hodgkinson  and  E.  G.  C.  Poole§. 

§  4-63.  The  mapping  function  for  a  rectangle.  When  n  —  4  and 
a  ^  p  ^  y  -=  8  =  ^,  the  polygon  is  a  rectangle  and  z  is  represented  by  an 
elliptic  integral  which  can  be  reduced  to  the  normal  form 

z  =  H  f  dt  [(1  -  t*)  (1  -  &2*2)]-i 
Jo 

by  a  transformation  of  type 

w(Ct  +  D)  -  At  +  B.  ......  (A) 

If,  in  fact,  the  integral  is 

dw  [(w  —p)  (w  —  q)  (w  —  r)  (w  —  s)]~%9 

we  have  (Ct  -f  D)  (w  -  p)  -  (A  -  Cp)  t  -f  B  -  Dp, 

(Ct  -h  D2)  dw  -  (AD  -  BC)  dt, 


*  Proc.  London  Math.  Soc.  (2),  vol.  xxn,  p.  337  (1924).  f  ^id.  p.  425. 

J  Ibid.  p.  389.  §  Ibid.  vol.  xxra,  p.  396  (1925). 


Mapping  of  a  Rectangle  303 

and  so  the  transformation  reduces  the  integral  to  the  normal  form  if 

A  -  Cp  =  B  -  Dp, 
A  -  Cq  =  Dq  -  B, 
A  -  Cr  =  k  (B  -  Dr), 
A  -  Gs  =  Ic  (Ds  -  B). 
These  equations  give 

C  (q  -  p)  =  2B  -  D  (p  4-  g), 
C  (s  -  r)  =  2kB  -  kD  (r  +  s), 
2A  -  (7  (p  +  <?)  =  JQ  (q  -  p), 
2A  -  C  (r  +  s)  =  kD  (s  -  r), 
C[s-r-  k(q-p)]=  kD[p  +  q-r-  s], 
C  [r  +  s  -  (p  +  q)]  -  D  [q  -  p  +  k  (r  -  a)], 

£2  (?  _  p)  (r  _   3)   +   k  [(q  _  p)2  +    (r  _   0)3  _    (p  +   ?  __   f  __  j)S] 

+  (?  ~  P)  (r  ~  s)  =  0. 
This  equation  gives  two  values  of  k  which  are  both  real  if 

[(?  ~  P?  +  (r  ~  «)2  -  (p  +  ?  -  r  -  «s)2]2  >  4  (gr  -  ^))2  (r  -  s)*, 
that  is,  if 

[(?  -  P  +  r  ~  s)2  ~  (P  +  ?  ~  r  -  5)21  [(?  ~  P  ~  r  +  <*)2  ~  (p  +  ?  -  r  -  5)2]  >  0, 
or,  if  4  (#  —  «s)  (r  —  p)  (q  —  r)  (s  —  p)  >  0. 

Itr^p^>q^s  this  is  evidently  true  and  since  the  product  of  the  two 
values  of  k  is  unity,  we  may  conclude  that  one  value  of  k  is  greater  than  1  , 
the  other  less  than  1.  This  latter  value  should  be  chosen  for  the  transfor- 
mation. With  this  value 


consequently,  the  transformation  (A)  transforms  the  upper  half  of  the 
w-plane  into  the  upper  half  of  the  £-plane. 

When  the  normal  form  of  the  integral  is  used  the  lengths  of  the  sides 
of  the  rectangle  are  a  and  b  respectively,  where 


a  -  H        dt  [(1  -  t2)  (1  -  W)]-*  -  2HK, 
J-i 

ri/fc 
6  =  ff  I      dt  [(1  -  t*)  (1  -  4V)]r*  =  HK', 

and  where  4X  and  2iK  are  the  periods  of  the  elliptic  function  sn  u  defined 
by  the  equation  x  =  sn  u,  where 

u^  \X  dt[(l  -  t*)  (1  -  tV)]-». 
Jo 

With  the  aid  of  this  function  £  can  be  expressed  in  the  form 

t  -  sn  (z/H). 


304  Conformal  Representation 

The  modulus  k  may  be  calculated  with  the  aid  of  Jacobi's  well-known 

r(\  -fg'Ml  +      )  (1  f. 


formula 


in  which  </  =  exp  [—  TrK'/K]  =  exp  [—  2?r6/a]. 

When  the  region  is  of  the  type  shown  in  Fig.  23  the  internal  angles  of 
the  polygon  are  877/2  at  four  corners  and  ?r/2  at  the  other  eight.  The 
transformation  is  thus  of  the  type 
2  -  A  [(w  -  cj  (w  -  c2)  (w  -  c3)  (w  -  c4)]*  [(w  -  fa)  ...  (w  -  pB)]~*dw  +  B. 

A  particular  transformation  of  this  type  is  obtained  by  assigning 
positive  values  of  w  to  corners  of  the  polygon  which  lie  above  the  axis  of 
x  and  negative  values  of  w  to  corners  which  lie  below  the  axis  of  x,  points 
which  are  images  of  each  other  in  the  axis  of  x  being  given  parameters 
whose  sum  is  zero.  The  transformation  is  now 


z  = 


Fig.  23.  Fig.  24. 

Making  the  parameters  ax,  a2  tend  to  zero  and  the  parameters  6lf  62 
tend  to  infinity,  the  transformation  becomes 


,  (w*  - 


B, 


and  the  interior  of  the  polygon  becomes  a  region  which  extends  to  infinity. 

To  use  this  transformation  for  the  solution  of  an  electrical  problem  in 
which  the  two  pole  pieces  in  Fig.  24  are  maintained  at  different  poten- 
tials,  we  write*  «,  =  ;CM*,  x  =  0  +  ^, 

so  as  to  map  the  half  of  the  w-plane  for  which  v  >  0  on  the  strip  —  TT  <  <f)  <  TT. 
This  will  make  w>  =  0  correspond  to  z  =  0  if  B  =  0,  and  the  lower  limit  of 
the  integral  is  i  \/c. 

Writing  ck  =  1  we  find  that  the  lengths  a  and  6  in  the  figure  are  given 

by  the  equations  re  ^ 

26=  <7c      ~a/(a), 

./i  5 

^    f1    d$  «  ,  .       ~    [~lds  j.  .  . 
-  2*a  =  Cc        -2/  (5)  -  Cc         -2  /  (a), 

Jt*/c  5  JiJc  s 

*  Riemann-Weber,  Differentialgleichungen  der  Physik,  Bd.  n,  S.  304. 


Region  outside  a  Polygon  305 

where  f  (s)  =  [(1  —  s2)  (1  —  k2s2)]*.  These  integrals  are  easily  reduced  to 
standard  forms  of  elliptic  integrals,  thus 
cds  -  c    ds 


ds  **     c  ds 


Now  if  we  put  ksr  =  1,  the  last  integral  becomes 

~  _  r  (L~JL?!L^ 

~    Ji  ~7F) 


and  we  eventually  find  that 

6  =  Cc[2Ef-  (I-  k*)K'], 
a-  2Cc[2E  -  (1  -  ia)  JL]. 
26  2^'  -  (1  -  A8)  #' 


Therefore 


a        2E-  (I  -  P)  K  ' 


EXAMPLE 
Prove  that  if  OABC  is  the  rectangle  with  sides  x  =  0,  x  =  K,  y  =  0,  y  =  K'  and 

^  -f  it/t  =  log  (an  z), 
we  have  ^  =  0  on  OA,  ABt  BC;  0  ==  rr/2  on  CO.  Prove  also  that  if 

^  4-  itj,  =  log  (en  z), 

where  (en  z)2  4-  (sn  z)2  =  1,  we  have  0  =  0  on  0.4,  0(7;  t/i  =  —  n/2  on  J5^4,  J5C. 
See  GreenhilTs  Elliptic  Functions,  ch.  ix. 

§  4-64.  Conformal  mapping  of  the  region  outside  a  polygon.  In  order  to 
map  the  region  outside  a  polygon  on  the  upper  half  of  the  w-plane,  we  may 
proceed  in  much  the  same  way  as  before,  but  we  must  now  use  the  external 
angles  of  the  polygon  and  must  consider  the  point  in  the  w-plane  which 
corresponds  to  points  at  infinity  in  the  z-plane.  Let  us  suppose  that  the 
w-plane  is  chosen  so  that  this  point  is  given  by  w  =  i,  then  there  should 
be  an  equation  of  the  form 

— ~—      C       C  (w  -  i) 
where  the  coefficients  Cm  are  constants.  This  gives 

_?  —  _  i   Q    i 

dw  (w  —  i)2         l       *"' 

d  ,       dz  2  -~ 

log  ~-  = .  -h  P  (w  —  ^), 

aw     °  aw  w  —  i 

where  P  (w  —  i)  is  a  power  series  in  w  —  i. 

Since  -3—  log  -j—  is  to  be  real  it  must  be  of  the  form 

dw    &  dw 

—  1      —  -  S  a-^~l  -      2 2 

dw     °  dw          w  —  a      w  —  i      w  +  i% 

B  20 


306  Conformal  Representation 

Therefore 

w-  a)"-1  (w  -  6)*-1  ...  (w  -  iy~l  (1  4-  w*)~*dw  +  C',  ...(I) 

where  C  and  C'  are  arbitrary  constants  of  integration.  The  relation  between 
the  indices  a  is  now 

S  (a  -  1)  =  2, 

for  the  sum  of  the  exterior  angles  of  a  polygon  with  n  vertices  is  (n  4-  2)  TT. 
The  region  outside  a  polygon  can  be  mapped  on  the  exterior  of  a  unit 
circle  with  the  aid  of  a  transformation  of  type 


-  a)*-1  (w  -  b)?-1  ...  w~2dw,     |  a  |  =  |  6  |  =  ...  =  1, 

where,  as  before, 

S  (a  -  1)  -  2. 

When  the  integrand  is  expanded  in  ascending  powers  of  w~l  there  will 
be  a  term  of  type  w1  which  will,  on  integration,  give  rise  to  a  logarithmic 
term  unless  the  condition 

Sa(a-  1)  =  0 
is  satisfied. 

When  the  polygon  has  only  two  vertices  and  reduces  to  a  rectilinear 
cut  of  finite  length  in  the  z-plane,  we  have  a  =  ft  =  2.  The  second  condition 
may  be  satisfied  by  assigning  the  values  w  —  ±  1  to  the  ends  of  the  cut. 
The  transformation  is  now 


=  H  f  (w2-  l)w-2dw, 


and  the  length  of  the  cut  evidently  depends  on  the  value  of  H.  Taking 
H  —  £  for  simplicity,  the  transformation  becomes 

2z  =  w  4-  w~l. 
This  is  the  transformation  discussed  in  §  4-73. 

The  general  theorem  (I)  indicates  that  the  regibn  outside  a  straight 
cut  may  be  mapped  on  the  upper  half  of  the  w-plane  by  means  of  the 
transformation 

ft  f    1  -  w2    ,  2w 

ty     O      I  fJIH    — 

Z  —   ^    I    71i      , oTo  ™w  —    -i      ,"       o« 

J  (1  4-  w2)2  1  4-  w2 

The  region  outside  a  cut  in  the  form  of  a  circular  arc  may  be  obtained 
from  the  region  outside  a  straight  cut  by  inversion.  If  the  arc  is  taken  to 
be  that  part  of  the  circle  z  =  —  ie*ie,  for  which  —  a  <  6  <  a,  the  trans- 
formation 

4-14-  2iw  tan  a 


z  =  — 


w2  4-  1  —  2iw  tan  a 


maps  the  region  outside  the  arc  on  a  half  plane. 

Suppose  that  in  the  w-pjane  there  is  an  electric  charge  at  the  point 
w  =  i  (sec  a  4-  tan  a)  =  is,  say,  and  that  the  real  axis  is  a  conductor. 


Semicircular  Arc  307 

The  corresponding  charge  in  the  z-plane  will  be  at  infinity  and  the  circular 
arc  will  be  a  conductor  which  must  be  charged  with  a  charge  of  the  same 
amount  but  of  opposite  sign.  The  solution  of  the  potential  problem  in  the 
w-plane  is  evidently 

i    ,    -  1       i      w  —  is 
y  =  0  -f  ii/f  =  log    ---  —  . 

*         V  -T-   "T  6  w  _j_  ls 

mu-      •  i.i_  /i    \  -  1  +  &~x  sin 

This  gives          ur  =  -  is  coth 


and  finally 

X=  —  log    ^  cosec  a  {z  4-  1  4-  (z2  4-  2iz  cos  2a  —  1)1}    . 

The  two-valued  function  (z2  +  2^z  cos  2a  —  l)i  may  be  regarded  as 
one-valued  in  the  region  outside  the  cut  and  must  be  defined  so  that  it  is 
equal  to  i  when  z  =  0  and  is  of  the  form  —  z  —  i  cos  2a  when  j  z  \  is  very 
large.  Changing  the  signs  of  <f>  and  x  we  have 

x  =  K  [2  sin  a  cosh  <f>  sin  0  -f  sin2  a  sin  20], 
^  =  _  jf  [l  -f-  2  sin  a  cosh  </>  cos  0  4-  sin2  a  cos  20], 
where  K~l  =  1  4-  2e~^  sin  a  cos  0  -f  e"2^  sin2  a. 


With  the  aid  of  these  equations  Bickley  has  drawn  the  equipotentials 
and  lines  of  force  for  the  case  of  a  semicircular  arc.  The  charge  resides  for 
the  most  part  on  the  outer  face,  the  surface  density  becoming  infinite  at 
the  edges.  The  field  appears  to  be  approximately  uniform  ori  the  axis  just 
above  the  centre  of  the  circle. 

The  field  at  a  great  distance  from  the  circular  arc  is  roughly  that  due 
to  an  equal  charge  at  the  point  z  =  —  i  cos2  «,  for  when  x  is  large,  the 
equation 

.   1  -f  e*  sin  a  .  r  ,  .       ._  _  , 

z  =  —  i  -.-  ------  --.  —  =  —  i  [1  -f  ex  sin  a]  [1  —  e~*  sin  a]  ... 

1  4-  e~x  sin  a  L  J 

may  be  written  in  the  form 

z  4-  i  cosa  a  =  —  iex  sin  a  4-  negligible  terms. 

This  point,  which  may  be  called  the  "  centre  of  charge,"  is  the  middle 
point  of  that  portion  of  the  central  radius  cut  off  by  the  chord  and 
the  arc. 

On  the  circular  arc 

X  =  i0  and  z  ==  —  ie2ld. 

Therefore  sin  (0  —  0)  sin  a  =  sin  9. 

The  surface  density  is  thus  proportional  to  S  cos  8  4-  1  on  the  convex 
face  and  to  S  cos  6  —  1  on  the  concave  face,  S  denoting  the  quantity 

S  =  (sin2  a  -  sin2  0)"*. 
If  E  is  the  charge  per  unit  length  of  a  cylindrical  conductor  whose 


308  Conformal  Representation 

cross-section  is  the  circular  arc  and  d  is  the  diameter  of  the  circle,  the 
surface  density  a  is  given  by  Love's  formula 

2-rrcrd  =  E  |  sec  v  \  (cosec  a  —  cos  v), 

where  sin  v  =  —  tan  0  cot  a. 

The  solution  of  the  electrical  problem  of  a  conducting  plate  under  the 
influence  of  a  line  charge  parallel  to  the  plate  but  not  in  its  plane  may  be 
derived  from  the  preceding  analysis  by  inversion  from  a  point  0  on  the 
unoccupied  part  of  the  circle.  Let  AB  be  the  cross-section  of  the  plate, 
/)'the  foot  of  the  perpendicular  from  O  on  AB,  OC1  the  bisector  of  the 
angle  AOB,  then  the  surface  density  cr  is  given  by  Love's  formula 

_  E  OD  cosec  a  —  cos  v 
°  =  277  OP2    "J cos ~v~\        ' 
where  now  sin  v  =  cot  a  tan  (P'OC'), 

cos  v  =  ±  cosec  a    — ^-p — -  , 

A'P'C'B'  being  perpendicular  to  OC1  (Fig.  25).  Thus 
__  E  OD  OA'  =f  (A'P' .  J5'P')_* 
07  ~  277  OP2         (A'P'.WP')* 

O 


Fig.  25. 

This  is  easily  converted  into  the  expression  given  in  §  3-81. 
The  region  outside  a  rectangle  may  be  mapped  on  the  interior  of  the 
unit  circle  in  the  £-plane  with  the  aid  of  the  transformation 


z=  f  ds  ( 1  -  2s2  cos  2a  + 
Ji 


while  a  transformation  which  maps  the  region  outside  the  rectangle  into 
the  region  outside  the  circle  is  obtained  by  using  a  minus  sign  in  front  of 
the  integral. 

Let  us  use  this  transformation  to  determine  the  drag  on  a  long  thin 
rod  of  rectangular  section  which  is  moved  slowly  parallel  to  its  length 
through  a  viscous  liquid  contained  in  a  wide  pipe  of  nearly  circular  section. 
We  write  log  £  =  iw  =  i  (u  -f  iv),  where  v  is  the  velocity  at  any  point  in 
the  z-plane,  then 


=  2*  f    (cos  2a  -  cos  2s)$  ds  =  2  f    (sin2  a  -  sin2  <$)*  ds. 
Jo  Jo 


Hydrodynamical  Applications  309 

Putting  sin  s  =  sin  a  sin  0,  this  becomes 


-  2  f  «?^lizJE^!)_f  -  2  I*  (1  -  sin*  «  si 
Jo    (1  -  sin2  a  sin2  j3)*  Jo 

-  2  f  cos2  a  (1  -  sin2  a  sin2  j8)~i 
Jo 


At  the  corner  A  immediately  to  the  right  of  the  origin  0  in  the  z-plane, 
we  have  6  =  |TT,  and 

XA  =  2E  (k)  -  2k'2  K  (k), 

where  k  =  sin  a  and  E  (k),  K  (k)  are  the  complete  elliptic  integrals  to 
modulus  k.  The  drag  on  the  half  side  OA  of  the  rectangle  is  proportional 
to  WA)  and  since 

sin  WA  —  sin  a  sin  (|TT)  =  sin  a, 

we  have  WA  =  a.  The  drag  on  the  side  OA  is  thus  equal  to  (a/2?r)  times  the 
drag  of  the  whole  rectangle.  [C.  H.  Lees,  Proc.  Roy*.  Soc.  A,  vol.  xcn, 
p.  144  (1916).] 

EXAMPLE 

A  line  charge  Q  at  the  origin  is  partly  shielded  by  a  cylindrical  shell  of  no  radial 
thickness  having  the  line  charge  for  its  axis,  the  trace  of  the  shell  on  the  #y-plane  being  that 
part  of  the  circular  arc  z  =  ae2lB  for  which  —  TT  <  —  2co  <  20  <  2o>  <  TT.  Prove  that  the 
potential  <f>  is  given  by  the  formula 

(z  ~  a)  cos2  ">  -f-  (z  -f  a)  sin2  a>  -f  R 


a)  8n  „  _  (z  _) 

where  J2  denotes  that  branch  of  the  radical  [z2  —  2az  cos  2aj  -f  a2]*,  whose  real  part  is 
positive  when  the  point  z  is  external  to  the  circle. 

The  surface  density  a  of  the  induced  charge  at  a  point  0  on  the  charged  arc  is 

a  =  _  Q  {Sec  a>  (tan2  o>  -  tan2  0)~*  ±  l}/47ra, 

the  upper  sign  corresponding  to  the  density  on  the  concave  side,  the  lower  sign  to  the  density 
on  the  convex  side.  The  latter  is  zero  when  2o>  =  -n-,  that  is,  when  the  circle  closes. 

[Chester  Snow,  Scientific  Papers  of  the  Bureau  of  Standards,  No.  642  (1926).] 

§  4-71.  Applications  of  conformal  representation  in  hydrodynamics. 
Consider  the  two-dimensional  flow  round  an  airplane  wing  whose  span  is 
so  great  that  the  hypothesis  of  two-dimensional  flow  is  useful.  Let  u,  v 
be  the  component  velocities,  p  the  pressure,  p  the  density,  L  the  lift  per  unit 
length  of  span,  D  the  drag  per  unit  length  and  M  the  moment  about  the 
origin  of  co-ordinates,  this  moment  being  also  per  unit  length.  These 
quantities  may  be  calculated  from  the  flux  of  momentum  across  a  very 
large  contour  C  which  completely  surrounds  the  aerofoil.  In  fact,  if  /,  ra 
are  the  direction  cosines  of  the  normal  to  the  element  ds,  we  have 

L  -f  iD  =  —  p  \  (v  -f  iu)  (ul  -f  vm)  ds—  \  p  (m  -f  il)  ds, 

M  =  —  p  \  (xv  —  yu)  (ul  +  vm)  ds  —  \  p  (xm  —  yl)  ds\ 


310  Conformal  Representation 

the  sign  of  M  is  such  that  a  diving  couple  is  regarded  as  positive.  The 
equations  may  be  rewritten  in  the  form 

L  +  iD  =  l/o  j  (v  -f  fw)2  dfe  -  I  [p  +  Jp  (^2  -f  v2)]  (m  -f  fZ)  ds, 
M  =  \p\  [(u2  -  v2)  (mx  4-  ly)  -f  2tw  (my  -  Zx)]  efo 


f  (ly  ~  mx)[p 


-f 


where  2  =  x  -f-  iy.  Now  when  the  motion  is  irrotational  outside  the  aerofoil 
the  quantity  p  -f  \p  (u2  -f  v2)  is  constant,  also 

(m  -f  iZ)  rfs  =  0,        (ly  —  mx)  ds  =  0, 

hence  L  +  iZ>  =  lp  \  (v  4-  ft*)2  rfz. 

Taking  the  contour  to  be  a  circle  of  radius  r,  we  have 

(y2  -  x2)]  ds/r 


f  [^2  -  ^2  -  2fwv]  [x2  -  y2  +  2ixy]  ds/ir 

(^  -h  iu)2zdz, 

where  the  symbol  R  is  used  to  denote  the  real  part  of  the  expression  which 
follows  it.  These  are  the  formulae  o^Blasius*  but  the  analysis  is  merely 
a  development  of  that  given  by  Kutta  and  Joukowsky. 

The  integrals  may  be  evaluated  with  the  aid  of  Cauchy's  theory  of 
residues  by  expanding  v  -f  iu  in  the  form 


When  the  region  outside  the  aerofoil  is  mapped  on  the  region  outside 
the  circle  |  z'  \  —  a  by  a  transformation  of  type 


the  flow  round  the  aerofoil  may  be  made  to  correspond  to  a  flow  round 
the  circle  by  using  the  same  complex  potential  x  in  each  case.  Now  for 
our  flow  round  the  circle  we  may  write 

®X          '-          /v*  tf  2/?'2  4-    IK 

e    a/z    + 


Zeits.f.  Math.  u.  Phys.  Bd.  Lvm,  S.  90  (1909),  Bd.  LIX,  S.  43  (1910). 


Region  outside  an  Aerofoil  311 

where  V,  a  and  K  are  constants,  therefore 

.  dv       .  dv  dz' 

v  +  iu  =  i  -~  =  i  -=£  .  —  - 

dz        dz   dz 

'e-*-  We 


i/c          KCa        inc*  U'      . 
-  -f  s-       ---  ~    ~ 

27T2 


If  2'  =  aelf*  is  the  point  of  stagnation  on  the  circle  which  maps  into  the 
trailing  edge  of  the  aerofoil,  we  have 

K  =  2-jraU'  sin  (a  -  /?), 
U  =  nCTe—, 
L  +  iZ>  -  KpU'ne-*  =  KpU  =  2<napUUr  sin  (a  -  /?). 


§  4-72.  jT/^e  mapping  of  a  wing  profile,  on  a  nearly  circular  curve.  For 
the  study  of  the  flow  of  an  inviscid  incompressible  fluid  round  an  aerofoil 
of  infinite  span,  it  is  \iseful  to  find  a  transformation  which  will  map  the 
region  outside  the  aerofoil  on  the  region  outside  a  curve  which  is  nearly 
circular. 

If  the  profile  has  a  sharp  point  at  the  trailing  edge  at  which  the  tangents 
to  the  upper  and  lower  parts  of  the  curve  meet  at  an  angle  a,  it  is  convenient 
to  make  use  of  a  transformation  of  type 


Z  +  KC 

where  cr  =  (2  —  K)  TT. 

If  a  circle  is  drawn  through  the  point  —  c  in  the  £-plane  so  that  it  just 
encloses  the  point  c,  cutting  the  line  (—  c,  c)  in  a  point  c  -f  d,  say,  where 
d  is  small,  this  circle  will  be  mapped  by  the  transformation  into  a  wing- 
shaped  curve  in  the  z-plane.  This  curve  closely  surrounds  the  lune  formed 
from  two  circular  arcs  meeting  at  an  angle  a  at  each  of  their  points  of 
junction,  z  ==  *c,  z  =  —  KC.  The  curve  actually  passes  through  the  point 
—  KC  and  has  the  same  tangents  there  as  the  lune  derived  from  a  circle  in 
the  £-plane  which  passes  through  the  points  (—  c,  c)  and  touches  the  former 
circle  at  the  point  —  c. 

If  we  start  with  the  profile  in  the  z-plane  and  wish  to  derive  from  it 
a  nearly  circular  curve  with  the  aid  of  a  transformation  of  this  type,  the 
rule  is  to  place  the  point  —  KC  at  the  trailing  edge  and  the  point  KC  inside 
the  contour  very  close  to  the  place  where  the  curvature  is  greatest*.  This 
rule  works  well  for  thin  aerofoils,  but  it  has  been  found  by  experience  that 
by  increasing  the  magnitude  of  d  an  aerofoil  with  a  thick  head  may  be 
obtained  from  a  circle,  and  that  the  thickness  of  the  middle  portion  of  the 
aerofoil  is  governed  partly  by  the  value  of  cr.  Hence  in  endeavouring  to 

*  F.  Hohndorf,  Zeits.  f.  ang.  Math.  u.  Mech.  Bd.  vi,  S.  265  (1926). 


312  Conformed  Representation 

map  a  thick  aerofoil  on  a  nearly  circular  curve  the  point  KC  may  be  taken 
at  an  appreciable  distance  from  the  boundary.  Another  point  to  be  noticed 
is  that  when  a  circle  is  transformed  into  an  aerofoil  by  means  of  the 
transformation  (A)  the  smaller  the  distance  of  the  centre  from  the  line 
(—  c,  c)  the  smaller  is  the  camber  of  the  corresponding  aerofoil  and  the  more 
symmetric  is  the  head.  The  point  KC,  moreover,  lies  very  nearly  on  the  line 
of  symmetry. 

The  actual  transformation  may  be  carried  out  graphically  with  the  aid 
of  two  corresponding  systems  of  circles  indicated  by  the  use  of  bipolar 
co-ordinates.  The  circles  in  one  plane  are  the  two  mutually  orthogonal 
coaxial  systems  having  the  points  (—  c,  c)  as  common  points  and  limiting 
points  respectively;  the  corresponding  circles  in  the  other  plane  for  two 
mutually  orthogonal  systems  having  the  points  (—  c,  c)  as  common  points 
and  limiting  points  respectively.  This  is  the  method  recommended  by 
K£rm&n  and  Trefftz.  Another  construction  recommended  by  Hohndorf 
depends  upon  the  substitutions 


by  which  the  transformation  may  be  written  in  the  form 


where 

The  plan  is  to  first  consider  the  transformation  from  z  to  £,  given  by 
the  equation 

,1  C,  ~~"   C          (Z  ~~~   KC\  * 

r  =  t*  or    f—  =  (   — — )  . 
£  -f-  c      \z  4-  KC)    • 

This  transformation  may  be  performed  graphically*  by  writing 

z0  =  z  +  KC,     £0  =  J  +  c, 
when  the  relation  becomes 


When  $  has  been  found  its  ^th  root  may  be  determined  graphically 
and  when  this  is  multiplied  by  $  the  value  of  r  is  obtained,  and  from  this 
£  is  easily  derived. 

Hohndorf  gives  a  table  of  values  of  rj  corresponding  to  different 
angles  a.  When  a  -=  4°,  77  =  89,  when  a  =  8°,  77  =  44,  and  when  a  =  10°, 

*  The  transformation  may  also  be  performed  graphically  by  writing  it  in  the  form 

•-5('*?) 

when  it  is  desired  to  pass  from  a  figure  in  the  (-plane  to  a  corresponding  figure  in  the  2-plane. 


Aerofoil  of  Small  Thickness  313 

rj  =  85.  When  the  transformation  (A)  is  expressed  by  means  of  series, 
the  results  are 

_,      /c2  -  1  c2      (/c4  -  5*2  +  4)  c4 

+  •••> 


r  l-/c2c2      (4/c4-  5*2+  l)c4 

r  —  ~    i    __    __  v  __  _  __  _  _  _i  _    _i_ 

^         ^       3       z  45z3  ^"" 

§  4-73.    Aerofoil  of  small  thickness*.    We  have  seen  that  the  trans- 
formation 

z'  =  z  +  a2/z 

maps  the  circle  (7  given  by  |  z  \  =  a  into  a  flat  plate  P'  extending  from 
z'  =  2a  to  z'  =  —  2a  and  back.  On  the  other  hand,  if  the  A's  are  small 
quantities  the  transformation 

£  =  s{l+    S   ^n(a/2)"} 

n-O 

maps  C  into  a  curve  F  differing  slightly  from  a  circle,  and  if  we  then  put 


F  maps  into  a  curve  II'  differing  slightly  from  a  flat  plate.  Now  for  a 
point  on  F 

£  =  a(l  +  r)eie, 

where  6  is  a  real  angle  and  r  a  real  quantity  which  is  small  ;  therefore  to  the 
first  order  in  r 

£  =  2a  (cos  0  +  ir  sin  0), 

and  so  £'  =  2a  cos  0,     rjf  =  2ar  sin  0. 

For  points  on  (7  and  P'  we  may  use  a  real  angle  co  and  write 

z  =  aeia>,     a?'  =  2a  cos  co,     y'  ==  0, 

then  (1  4-  r)  e«»--)  =  1  +   S  ^«e-'»-, 

n»0 

i 

and  since  0  —  o>  is  small  we  have  to  the  first  order,  with  An  =  Bn  +  iCn  , 

r  =  S  (J5n  cos  ^^6o  -f  <7n  sin  7^0), 
0  —  </>  =  2  (Cn  cos  nco  —  -Bn  sin  /io>). 
Hence  by  Fourier's  theorem 

D     r        AJQ    (*  ^  c°s  n®  ^ 

7r.Bn  =        r  cos  n0d6  =        -±-  —.  —  w  dO, 


*  H.  Jeffreys,  Proc.  Roy.  Soc.  A,  vol.  cxxi,  p.  22  (1928). 


314  Conformal  Representation 

Since  sin  0  and  sin  n0  are  odd  functions  of  0,  whilst  cos  nO  is  an  *even 
function  of  6,  it  appears  that  Cn  depends  on  the  sums  and  Bn  on  the 
differences  of  the  values  of  rj'  corresponding  to  angles  ±  #;  thus  the  (7n's 
depend  on  the  camber  of  the  aerofoil,  the  J?n's  on  its  thickness. 

When  6  is  small,  that  is,  for  points  near  the  trailing  edge  of  the  aerofoil, 
we  have  approximately 

v)        _     r  sin  0    __  2r 
2a~~-~?  ^  r~"cos  0  =  ~0  ' 

and  when  TT  —  0  is  a  small  quantity  co  we  have 


Thus  r  vanishes  at  0  =  0  because  the  slope  of  the  section  is  finite  there  ; 
but  at  0  =  TT  the  section  and  the  axis  meet  at  right  angles  at  a  point  which 
may  be  called  the  leading  edge.  If  the  curvature  at  this  point  is  l/R  we 
have  to  a  close  approximation 

nr»          V2  4a2r2sin2#         _     .  .,  m        .     9 

2jR  =  ,.—  -s-  =  :r~  ,-  ---  -„-,  =  2ar2  (1  —  cos  5)  =  4ar2, 

^  +  2a      2a  (1  4-  cos  0)  v  ; 

consequently  r  is  finite  and  equal  to  (R/2a)%  at  the  leading  edge.  If  at 
a  great  distance  from  the  circle  C  the  flow  in  the  z-plane  is  represented 
approximately  by  a  velocity  U  making  an  angle  a  with  the  axis  of  x,  we 
have 

IK 
x  =  </,  +  ie/r  =  C7ze-*  -f  Ue^at/z  +  ^  log  (z/a), 

where  K  is  the  circulation  round  the  cylinder.  Taking  x  to  be  the  complex 
potential  for  a  corresponding  flow  in  the  £'-plane  the  component  velocities 
(u9  v)  in  this  plane  are  given  by  the  equation 


To  determine  K  we  make  the  velocity  finite  at  the  trailing  edge  where 
f '  is  a  maximum  and  0  =  0,  r  =  0,  £  =  a,  -^ r  =  0. 
Hence  d^/dz  is  zero  and  so 

But  when  0=0 


say,  where  j8  ==  S  Cn . 

Therefore  /c  =  ±naU  sin  (a  + 


Thin  Aerofoil  315 

Now 


{1-2  '(n  -  1MM  (a/zf«}"(l  -  a'/T2) 

«  1  '  (1  +  fl>  +  41o/g')/2"£'  , 

" 


+ IK 


Therefore  by  the  Kutta-Joukowsky  theorem  the  lift  per  unit  span  of 
the  aerofoil  is 


L  -    -  =  477/>a  (1  +  50)  sin  (a  +  jB)  F2,     7(1  +  J50)  =  J7, 

1  H~  -^0 

and  the  lift  coefficient  is 

#L  =  (L/4paV*)  =  77  (1  +  J?0)  sin  («  +  )8). 

The  thickness  thus  affects  the  lift  through  J30,  which  is  a  positive 
constant  for  a  given  wing. 

The  moment  about  0,  that  is  a  point  midway  between  the  leading  and 
trailing  edges  of  the  aerofoil,  is  equal  to  np  times  the  real  part  of  the 

coefficient  of  -f  i£'~2  in  the  expansion  of  (-=£  j  .  This  coefficient  is 


and  so  to  the  first  order  in  the  J3's  and  C"s  the  moment  is  M  ,  where 
M  -  Z-n-pVW  {C2  cos  2a  -  (1  +  B2)  sin  2a  f  2Bl  cos  a  sin  (a  +  ]8) 

4-  2^  sin  a  sin  (a  +  ]3)}. 
The  moment  about  the  leading  edge  is 


where  terms  of  orders  a2,  aBn>  aCn  have  been  retained,  but  terms  of  orders 
a3,  a2J?n,  a2Cn,  dropped.  When  squares  and  products  of  the  JB's  and  C"s 
are  neglected,  the  moment  coefficient  KM  is 


=  \KL  (1  +  B.  +  B,- 
The  moment  coefficient  at  zero  lift  is  thus 


and  is  independent  of  the  thickness  to  this  order  of  approximation.  The 
thickness,  however,  affects  the  coefficient  of  KL. 


316  Conformal  Representation 

For  further  applications  of  conformal  representation  in  hydrodynamics 
the  reader  is  referred  to  H.  Glauert's  Aerofoil  and  Airscrew  Theory  (Cam- 
bridge, 1926)  and  to  H.  Villat's  Lemons  sur  VHydrodynamique  (Gauthier- 
Villars,  Paris,  1929). 

§  4«  81.  Orthogonal  polynomials  associated  with  a  given  curve*.  Let/  (z) 
be  a  function  which  is  defined  for  points  of  the  z-plane  which  lie  on  a  closed 
continuous  rectifiable  curve  C  which  is  free  from  double  points.  If 

ds  =  I  dz  I  ,  the  integral  r 

f(z)ds 

J  C 

denotes  as  usual  the  limiting  value 


lim     2 

m->co  f»»l 

where  z0  ,  zl  ,  z2-5  .  .  .  represent  successive  points  on  C  for  which 

lim   [Maximum  value  of  |  zv  —  zv_\  \  for  0  <  v  <  m]  —  0,     (zl  =  zm), 

in  —  >•  oo 

and  £„  denotes  an  arbitrary  point  of  C  which  lies  between  zv^  and  zv.   We 
have  in  particular  /• 

ds  =  l, 

Jc 

where  I  denotes  the  length  of  the  curve  C.  We  shall  suppose  now  that  the 
unit  of  length  is  chosen  so  that  I  =  1  . 

Using  z  to  denote  the  complex  quantity  conjugate  to  z,  we  write 


D0  =sAoo  = 


n. 


Let  Hmn  denote  the  co-factor  of  hmn  in  the  determinant  Dn  ,  and  let  an 
be  a  constant  whose  value  will  be  determined  later  ;  then,  if 

Pn  (z)  =  an  (H0n  +  zHln  +  ...  zn  Hnn), 
it  is  easily  seen  that 

f    Pn  (z)  zvds  -  an  (hQvH0n  +  hlt,Hln  -f  ...  hn¥Hnn) 
Jc 


-  an  Dn  for  v  -  n, 
Pn  (z)  |2  dz  =  ananHnnDn  -  an 


*  See  a  remarkable  paper  by  Szego,  Jtfa^.  Zette.  Bd.  rx,  S.  218  (1921). 


Orthogonal  Polynomials  317 

The  polynomials  thus  form  an  orthogonal  system  which  is  normalised 
by  choosing  an,  so  that  an  =  an  =  (A»  Ai-i)"*- 

If  Pm  (z)  denotes  the  complex  quantity  conjugate  to  Pm  (z),  the  ortho- 
gonal relations  may  be  written  in  the  form 


Pn  (z)  Pm  (z)  ds=  0,     m*n 
Jc 

=  1,     m  =  n. 

Let  us  now  suppose  that  C  is  an  analytic  curve  and  that  £  =  y  (z)  is 
the  function  which  maps  the  interior  of  C  smoothly  on  the  region  |  £  |  <  1 
of  the  £-plane  in  such  a  way  that  y  (a)  =  0,  y  (a)  >  0.  Since  C  is  an 
analytic  curve  y  (z)  is  also  regular  and  smooth  in  a  region  enclosing  the 
curve  C.  It  is  known,  moreover,  that  there  is  one  and  only  one  function, 
z  =  g  (£),  which  is  regular  and  smooth  for  |  £  |  <  1  and  maps  the  interior 
of  |  £  |  =  1  on  the  interior  of  the  curve  C.  The  derivatives  of  the  functions 
y  (z),  g  (£)  are  connected  by  the  relation  y  (z)  g'  (£)  =  1,  where  z  and  £  are 
associated  points  of  the  two  planes. 

Our  aim  now  is  to  show  that 


o  rz 

=   lim    ~-,J-r      [Kn(a,w)]*dw, 

n_>oo  J^n  Va>  a)  J  a 


where  Kn  (a,  z)  =  P0  (a)  P0  (z)  +  ...  Pn  (a)  Pn  (z). 

We  shall  first  of  all  prove  an  important  property  of  the  polynomial 
Kn  (a,  z). 

Let  a  be  arbitrary  and  Gn  (z)  a  polynomial  of  the  nih  degree  with  the 
property 

f    \Gn(z)\*ds  =  1, 
Jc 

then  the  maximum  value  of  |  Gn  (a)  \ 2  is  Kn  (a,  a),  and  this  value  is  attained 
when  Gn  (z)  =  eKn  (a,  z)  [Jf  n  (a,  a)]~~i,  where  e  is  an  arbitrary  constant 
such  that  |  e  |  =  1. 
Let  us  write 

Gn  (z)  =  ^0P0  (z)  +  ^P!  (z)  +  ...  tnPn  (z), 

where  the  coefficient  tv  is  determined  by  Fourier's  rule  and  is 


=  f 

J 


We  then  have 

(z)\*ds=    |*0|*-f    |^|2+   ...    \tn\\ 


=    f 
J 


C 

G  (a)  =  /0P0  (a)  +  ^P!  (a)  +  ...  ^nPn  (a), 
and  by  Schwarz's  inequality 


318  Conformal  Representation 

The  sign  of  equality  can  be  used  when 

t,  =  e!\W[Kn(a,a)]-*-, 
that  is,  when  On  (z)  =  eKn  (a,  z)  [Kn  (a,  a)]~i. 

When  the  point  a  is  within  the  region  bounded  by  Cy  and  F  (z)  is  any 
function  which  is  regular  and  analytic  in  the  closed  inner  realm  of  (7,  we 
have  the  inequality 


where  8  is  the  least  distance  of  the  point  a  from  the  curve  C.  To  prove 
this  we  remark  that  Cauchy's  theorem  gives 


and  so  |  F  (a)  \  <  ~  f     ~-(z)   ds  <  ~  [    \F  (z)  I  da. 

'  '       2n  Jc  z  —  a  277-8  Jp  '  ' 

In  the  special  case  when  F  (z)  =  \Gn  (z)]2  the  inequality  gives 


This  is  true  for  all  polynomials  Gn  (z),  and  so,  in  particular, 

27r8 
Since  8  is  independent  of  n,  this  inequality  establishes  the  convergence 

of  the  series 

K(a,a)=  |P0(a)|2  +  |  Pl  (a)  |2  +  ... 

for  the  case  in  which  the  point  a  lies  in  the  region  bounded  by  C.  Again, 
we  have  the  inequality 

\Kn  (a, 2)  |2<  [|P0  (a)  ||  Po  (z)  |  +  |  Pl  (a)  \\  P,(z)  \  +  ...]»<  Kn  (a, a)  Kn  (z,z), 
and  if  Rnm  (a,  z)  denotes  the  remainder 

Rnm  (a,  z)  =  P^TF)  Pn+l  (z)  4-  ...  Pn+m  (a)  Pn+m  (z), 
we  have  |  Rnm  (a,  z)  |2  <  Rnm  (a,  a)  Rnm  (z,  z). 

Since  the  series  K  (a,  a)  is  convergent  when  a  lies  within  C  we  can  find 
a  number  N  (a)  such  that  if  n  >  N  (a)  we  have  for  all  values  of  m 

|  Rnm  (a,  a)  |  <  e, 

where  e  is  a  small  positive  quantity  given  in  advance,  hence  if  N  is  the 
greater  of  the  two  quantities  N  (a),  N  (z),  we  have  for  n  >  N 

This  establishes  the  convergence  of  the  series 


K  (a,  z)  -  KM  Pft  (z)  +  Pn(a)  P,  (z)  +  .... 


Series  of  Polynomials  319 

To  prove  that  the  series  is  uniformly  convergent  in  any  closed  realm  R 
lying  entirely  within  C  we  note  that  the  quantities  Kn  (a,  z)  are  uniformly 
bounded  in  the  sense  that 

|  Kn  (a,  z)  |2  <  Kn  (a,  a)  Kn  (z,  z)  < 


The  general  selection  theorem  of  §  4-45  now  tells  us  that  from  the 
sequence  Kn  (a,  z)  we  may  select  a  partial  sequence  of  functions  which 
converges  uniformly  in  R  towards  a  limit  function/  (z).  Since,  however, 
the  sequence  converges  to  K  (a,  z)  this  limit  function/  (z)  must  be  identical 
with  K  (a,  z)  and  so  the  series  which  represents  K  (a,  z)  converges  uniformly 
in  R,  a  and  z  being  points  within  R. 

We  now  consider  the  integral 


=  [ 

J 


Kn(a,z)-X{7'(z)}l\*ds, 
c 

where  A  is  a  constant  which  is  at  our  disposal.  We  have 

Kn(a,z)  \2ds  =  Kn(a,a), 


\    \y'(z)\ds=  |2"d0=2,r, 

JC  JO 


Kn  (a,  z)  {/  (z)}*  ds  =       Kn  {a,  g 


o 


I          V      f  ^f 

— •      I  J^        <  d 

where  £  =  e".  Jo 

Now  the  function  Kn  {a,  g  (£)}  vV  (0   is  regular  and  analytic   for 
|  £  |  <  1,  and  so  the  last  integral  is  equal  to 


n  {a,  g  (0)}  V)  =  ^Kn  (a,  a)  [/  (a)]-. 
Choosing  A  -  ~-  [y  (a)]*, 

we  have  finally  Jn  =  ^-  |  /  (a)  |  —  ^Tn  (a,  a). 

ZTT 

Our  object  now  is  to  show  that 

lim  Jn  -  0. 

n->oo 

Let  us  write  £(£)  =  fo' (£)]*• 

Since  gr'  (£)  ^  0  for  |  £  |  <  1,  a  branch  of  L  (£)  is  a  regular  analytic 
function  for  |  f  |  <  1. 

We  now  consider  the  set  of  analytic  functions  E  (£)  regular  in  |  £  |  <  1 
and  such  that  f2ir 

|L(£)tf(0|»<M=l. 
Jo 


320  Conformal  Representation 

Let  a  be  a  fixed  number  whose  modulus  is  less  than  unity,  then  the 
maximum  value  of  |  E  (a)  \  2  is 

[1-  |«|2]-i|L(«)|-2. 
To  see  this  we  put 

L(QE(l)  =  t0  +  tlt+...+tnF+...9 
then  on  the  above  supposition 


and  Schwarz's  inequality  gives 

\E(a)\*\L(a)\*<    £    \tn 


The  sign  of  equality  holds  when,  and  only  when, 
tn  =  ca",  n=0,l,  2,  .... 

that  is,  when  L  (£)  #  (£)  =  j-^^g. 

M  f2'        ^         _        271L 

w  Jo  1  1  ~s£i»~  r-i«i2> 

therefore  27r  |  c  |2  =  1  -  |  a  |2, 

and  so  JB(0-.(1-* 


-.-—  -. 

1    -    Jo          (27T)*Zf(C) 

Now  let  £  (0  =  -E  {y  (2)}  =  (?  (z)  =  (?  {g  (J)}, 

then  jE?  (^)  and  (7  (2)  are  simultaneously  regular,  and 

1=  fa'|L(0^(0|lde=  [8"|i7'(OII^U)|'de=  f  IGW*. 

Jo  Jo  Jc 

Finally,  E  (0)  -  (7  (a), 

so  that  max  |  E  (0)  |2  -  max  |  G  (a)  |2, 

therefore 

K  (a,  a)  =  max  |  G  (a)  |-  =  max  |  JB  (0)  |«  =  ^-/^  =  ^^  |  , 

and  so 

Km  -/„  =  ^  |  /  («)  |  -  ^  (a,  a)  =  ~  {|  /  (a)  |  -  |  g'  (0)  |}  =  0. 

n  ->  oo  ^^  ^7r 


Since  [^n  (a,  z)  -  A  {y'  (z)}Jp  =  Fn  (z) 

is  a  regular  analytic  function  in  the  closed  inner  realm  of  (7,  we  have  for 
any  point  z0  within  C  whose  least  distance  from  C  is  8, 


and  so  as  n  ->oo,  lim   |  .Pn  (z0)  |  =  0. 


Region  Outside,  a  Closed  Curve  321 


Hence  K  (a,  z,)  =   lim  Kn  (a,  z0)  =  A  {/ 

n->oo 

Furthermore,  since 

K  (a,  a)  -  j^  |  /  (« 

we  have  /  W  -  2.  I* 

o  rg 

and  so  y  (2)  =     ~     -r      [K  (a,  z0)]2  dzQ. 

A  (a,  a)  Ja 

If  the  curve  (7  instead  of  being  of  unit  length  is  of  length  I  the  ortho- 
gonal polynomials  Pn  (z)  are  defined  so  that 


and  the  general  formula  for  the  mapping  function  becomes 


where  e  is  a  number  with  unit  modulus  and  is  equal  to  unity  when  the 
mapping  function  is  required  to  be  such  that  y'  (a)  >  0. 

A  study  of  the  expansion  of  functions  in  series  of  the  orthogonal  poly- 
nomials Pn  (z)  has  been  made  recently  by  Szego  and  by  V.  Smirnoff, 
Comptes  Rendus,  t.  CLXXXVI,  p.  21  (1928). 

§  4-82.  The  mapping  of  the  region  outside  C'  .   If  we  write 
z'  (z  —  a)  =  1,     ww'  =  1, 

the  interior  of  C  maps  into  the  region  outside  a  closed  curve  C'  in  such 
a  way  that  the  point  z  =  a  maps  into  the  point  at  infinity  in  the  z'-plane. 
The  interior  of  the  unit  circle  |  w  \  <  1  is  likewise  mapped  into  the  region 
|  w'  |  >  1,  the  point  w  =  0  corresponding  to  w'  =  oo. 

Hence  the  region  |  wr  \  >  1  is  mapped  on  the  region  outside  C'  in  such 
a  way  that  w'  =  oo  corresponds  to  z'  =  oo,  the  relation  between  the 
variables  being  of  type 

z'  =  rw'  +  T0  -f  T!  (w')~l  +  ...  +  rn  (w/)~n  -f  .... 
Since  w  =  y  (z),  the  function  which  gives  the  conf  ormal  representation  is 

w'  =  [y  {a  4-  (z')-1}]-1  =  0  (A  say. 

Szego  has  shown  that  the  function  0  (z')  may  also  be  obtained  directly 
with  the  aid  of  an  orthogonal  system  of  polynomials  Yln  (z')  associated 
with  the  curve  C". 

If  T  =  |  T  |  6fa,  we  have  in  fact  the  formula 


322  Conformal  Representation 

EXAMPLES 

1.   If  z0  is  a  root  of  the  equation  Pn  (z)  =  0,  prove  that 


—^--- I  zas. 

Z  -Z0| 

Hence  show  that  z0  lies  within  the  smallest  convex  closed  realm  R  which  contains  the 
curve  C. 

2.  If  C  is  a  circle  of  unit  radius, 

P  (z}  —  zn 

L  n  \*f  "~  *  > 


3.  If  the  curve  C  is  a  double  line  joining  the  points  —  1,1,  the  polynomial  P  (z)  becomes 
proportional  to  the  Legendre  polynomial.   Note  that  in  this  case  the  series  K  (a,  z)  fails  to 
converge,  but  this  does  not  contradict  the  general  convergence  theorem  because  now  the 
points  a  and  z  do  not  lie  within  C. 

4.  If  jRn  (z)  is  any  polynomial  and  a  any  point  within  the  curve  (7,  prove  that 


§  4-91.  Approximation  to  the  mapping  function  by  means  of  polynomials 
Let  a  circle  of  radius  R  be  drawn  round  the  origin  in  the  z-plane  and  let 

w  =  /  (z)  =  ao  +  aiz  +  a2z*  +  ••• 

be  a  power  series  converging  uniformly  in  its  whole  interior.  This  maps 
the  circle  on  a  region  of  the  complex  w-plane.  For  the  area  of  this  region 
we  easily  find  the  expressions 

A  =  I"  [2?r  I  /'  (z)  \*rdrde  (z  -  re«) 

Jo  Jo 


=  277  \R 

Jo 


rdr  S    |  an 

n=l 

al  I2  +  TT  S  n\  an  |2  R2n  +  ...  . 

n-2 

The  area  of  the  image  region  is  always  greater  than  77.R2  |  ax  |2  when 
ax  7^=  0  and  is  always  greater  than  TTH  |  an  |2J?2n  when  an  ^  0.  When  the 
mapping  function/  (z)  is  such  that  a:  =  1  the  result  is  that  the  area  of  the 
picture  is  greater  than  that  of  the  original  region  unless  the  picture  happens 
to  be  a  circle  of  radius  R. 

Suppose  now  that  we  are  given  a  simply  connected  smooth  limited 
region  B  of  the  z-plane.  Let  dr  be  an  element  of  area  of  this  region,  we 
then  look  for  a  function  /  (z)  regular  in  B  which  makes  the  integral 

7  =JJ   |/'(S)   |«(*T 

*  L.  Bieberbach,  Rend.  Palermo,  vol.  xxxvm,  p.  98  (1914). 


Approximation  by  Means  of  Polynomials  323 

as  small  as  possible.  To  make  the  problem  definite  we  add  the  restrictions 
that  /  (0)  =  0,  /'  (0)  =  1,  and  that  /  (z)  is  a  polynomial  of  the  nth  degree. 
These  conditions  are  satisfied  by  writing 

f(z)  =  z  +  a2z*+  ...+  anzn.  ......  (A) 

If  /  (z)  -f  €g  (z)  is  a  comparison  function  we  have  to  formulate  the 
conditions  that  the  integral 


(0  =  j]  I  /' 


+  *g'  (z)  \2dr^       [/'  (z)  +  eg'  (z)]  [T(z)  +  eY~&}  dr 


may  be  a  minimum  for  €  =  0.  These  conditions  are 
~ 


=    g'  {z)  *rw  dr  =     I  v'  M  I2  dr- 


The  inequality  is  always  satisfied,  but  the  two  equations  are  satisfied 
for  all  forms  of  the  polynomial  g  (z)  only  when  the  coefficients  as  satisfy 
certain  linear  equations.  If 


where  z  is  the  conjugate  of  z,  these  equations  are 
2z0>1  4-  4zltla2  +  6z2jla3  4-  ...  2nzn_lt  lan  =  0, 


i  +  (n.%)  Zi,n~ia*  +  (n.'3)  z^^a^  +  ...  (n.n)  zn_^n^an  -  0, 
and 

2zlf04-  (2.2)z1>1a2-f  (2.3)z1,2a3+  ...  (2.n)zltW.1oB  =  0, 

^n-ifo  +  (ra.2)  zn_1>]La2  -f  (w.3)  2;n_1>2a3  +  ...  (n.n)  zn_lin^dn  =  0. 
These  linear  equations  are  associated  with  the  Hermitian  form 

n2ln2    (p+  1)  (5+  l)^a^, 

p=0   g«0 

and  possess  a  single  set  of  solutions  for  which  /  is  a  minimum.   By  giving 
different  values  to  n  we  obtain  a  sequence  of  polynomials  which  in  many 
cases  converges  towards  a  limit  function  F  (z).  The  question  to  be  settled 
is  whether  this  function  F  (z),  among  all  mapping  func-       ^      -\ 
tions  with  the  properties  F  (0)  =  0,  F'  (0)  =  1,  gives  the     f  \ 

smallest  possible  area  to  the  picture  into  which  B  is    I  ^     \ 

mapped.  The  following  simple  example  tells  us  that  this 

is  not  always  the  case.   Consider  the  region  B  which 

arises  from  a  circle  when  the  outer  half  of  xme  of  its 

radii  is  added  to  the  boundary  (Fig.  26).    There  is  no  Fi&-  26' 

polynomial  which  maps  this  region  B  on  a  region  of  smaller  area.    For 


324  Conformal  Representation 

by  means  of  a  polynomial  the  region  B  is  mapped  on  another  region 
which  has  the  same  area  as  the  region  on  which  the  complete  circle  is 
mapped  and,  unless  the  polynomial  is  simply  z,  this  region  has  an  area 
which  is  greater  than  that  of  the  circle.  Hence  in  this  case  all  minimal 
polynomials  are  equal  to  z  and  F  (z)  is  also  equal  to  z. 

Bieberbach  has  investigated  the  convergence  of  the  sequence  of 
polynomials  to  the  desired  mapping  function  for  the  type  of  region 
discovered  by  Carath^odory*.  For  such  a  region  the  boundary  is  contained 
in  the  boundary  of  another  region  which  has  rib  point  in  common  with  the 
first.  The  interior  of  a  polygon  is  a  particular  region  of  this  type  and  so 
also  is  the  interior  of  a  Jordan  curve. 

Bieberbach's  method  of  approximation  has  been  used  recently  in 
aerofoil  theory  for  the  mapping  of  a  circle  on  a  region  which  is  nearly 
circular  f. 

Introducing  polar  co-ordinates,  z  =  reie,  and  supposing  that  on  the 
boundary  r  =  1  -f  y,  where  y  is  small,  we  may  write 

T2TT  fl+y  I  r2ir 

m      }Q        Jo  2>  4-  <7  +  2  J  o 

Hence  retaining  only  terms  up  to  the  second  order  in  the  binomial 
expansion  of  (1  +  y)p+<J+2? 

ZPQ=  \     y-cos  (p  -  q)  d.dO  +  P   -f  y2.cos  (p  -  q)  0.d0 

Jo  *         J  o 


(p  -  q)  d.dd  +     --  2.sin  (p  -  q)  0.d0,     p 


These  quantities  may  be  determined  from  the  profile  of  the  nearly 
circular  curve  when  this  is  given. 

Now  writing         zw  -  fw  +  irj^,     ap  =  <f>p  +  i$p, 

where  fOT,  77^,  <f>p  and  ifjp  are  all  real,  and  neglecting  all  the  coefficients 
after  a4  in  the  expansion  (A),  we  obtain  the  following  equations  for  the 
determination  of  </>2  ,  </>3  ,  </>4  ,  i/j2  ,  03  ,  i/r4  : 

£01  +  2£ii</»2  +  3£12(/>3  -f  3^^  -h  4|13</>4  -f  47?13</r4  =  0, 

^w  +  2fu^a  -  3^12^3  -f  3|12</r3  -  47?13^4  +  4^1304  =  0, 

^02  +  2^12^2  ~  2^^  +  3f22S63  -f  4£23<£4  +  4^23^  -  0, 

??02  +   2^12^2  +    2^12^2  +   3f22</T3  ~    47723(^4  +   4^23^  -    0, 


%j  +  27713^2  +  2£13</r2  -f  37723^63  +  3^^  -f  4fa 

*  Jtfo^.  ^Inn.  vol.  LXXII,  p.  107  (1912). 

t  F.  Hohndorf,  Zeite.  f.  any.  Math.  u.  Mech.  Bd.  vi,  S.  265  (1926).  The  conf  ormal  representation 
of  a  region  which  is  nearly  circular  is  discussed  in  a  very  general  way  by  L.  Bieberbach,  Sitzungsber. 
der  preussischen  Akademie  der  Wissenschajten,  S.  181  (1924). 


DanielVs  Orthogonal  Potentials  325 

Eliminating  04  and  </r4  we  obtain  the  equations 
(0133)  +  2  (1133)  fa  +3  (1233)  fa  4  3  [1233]  </»3  =  0, 

[0133]  +  2  (1133)  fa  -  3  [1233]  <£3  +  3  (1233)  ^3  -  0, 

(0233)  +  2  (1233)  fa  -  2  [1233]  02  +  3  (2233)  <£3  -  0, 

[0233]  +  2  [1233]  </>2  4  2  (1233)  02  4-3  (2233)  </r3  -  0, 

where 

(jpgr*)  -  $„$„  -  ^  fw  -  rjprTjqs,     [pqrs]  =  17^17,.,  4  f^ifc,  -  i?prfw, 
and  these  finally  give  the  values 

vNfa  =  Z2,_3 ,     yAty,  -  Z2,_, ,         v  =  2,  3,  4, 
where  N  =  (1233)2  4-  [1233]2  -  (1133)  (2233), 

Zl  =  (0133)  (2233)  -  (0233)  (1233)  -  [0233]  [1233], 
Z2  -  [0133] (2233)  4  (0233) [1233]  -  [0233] (1233), 
ZB  -  [0133] [1233]  4  (0233) (1133)  -  (0133) (1233), 
Z4  -  [0233] (1133)  -  (0133) [1233]  -  [0133] (1233). 

§  4- 92.    DanielVs  orttiogonal  potentials.  Consider  a  set  of  polynomials 
pQ  (z),  Pt  (z),  ...  defined  by  the  equations* 

(0,0)  (1,0)  (71,0) 

(0,  1)  (1,  1)  (n,  1) 


where 


(0,  71-1)     (1,73 
1 

(0,0)     (0,1)     (0,71) 
(1,0)     (1,1)     (l,w) 


(71,71- 

zn 


and 


(n,0)    (TI,!)     (7i,  n) 
(m,  n)  =  ^-      zmzndr, 

the  integral  being  taken  over  the  region  to  be  mapped  on  a  unit  circle. 
A  denotes  here  the  area  of  the  region  and  dr  an  element  of  area  enclosing 
the  point  z.  These  polynomials  satisfy  the  orthogonal  conditions 

\  r  f  _ 

3  Jj  Prn  (z)  Pn  (*)  dr  -  0,     m^n 

=  1,     m  =  n. 

A   mapping   function  /  (z)   which   satisfies   the   conditions  /  (a)  =  0, 
/'  (a)  =  1,  is  given  formally!  by  the  expansion 


4 


W  f 

J  a 


Pl  (z') 


where 


fif  =  p0(a)  p0  (a)  + 


±  (a)  4  .... 


*  These  equations  are  analogous  to  those  used  by  Szego. 

f  This  series  does  not  always  represent  an  appropriate  mapping  function  as  may  be  seen  from 
a  consideration  of  the  circular  region  with  a  cut  extending  half-way  along  a  radius  as  in  §  4-91. 


326  Conformal  Representation 

To  see  this  we  write 

/'  (z)  =  flo^o  (z)  +  aiPi  (z)  4-  a2p2  (z)  4-  ... , 


(z) 

where  a0  ,  ax  ,  .  .  .  ;  a0  ,  ax  ,  ...  are  coefficients  to  be  determined,  so  that  the 
integral 


f  (z)f'(z).dr  =  a0a0  +  a,at  -f  ... 
may  be  a  minimum  subject  to  the  conditions 


/'(a)=l,    />)=!.  ......  (A) 

Differentiating  with  respect  to  a0,  al9  ...  ;  a0,  al9  ...  in  turn  we  find  that 

«n  =  &Pn  («),       W  =   0,   1, 


where  i  and  ^  are  Lagrangian  multipliers  to  be  determined  by  means  of 
the  equations  (A).   We  easily  find  that 

1  =  kS,     kS  =  1, 
and  so 

Sf  (z)  =  Po(a)  Po  (z)  +  ^1S)  Pl  (2;)  +  ...  . 

If  pn  (z)  =  un  ~  ivn  ,  where  un  and  vn  are  real  potentials  which  can  be 
derived  from  a  potential  function  <f>n  by  means  of  the  equations 

_  d<f,n  ty, 

M"  ~   dx  '      "  ~   dy  ' 

we  have  U[  (-^  ^  +  ^  ^  dr  =  0,     m^« 

J[  J  J  \  3x    3o;         dy    dy  / 

=  1,     m  =  7i. 

The  potentials  </>0,  </>1?  ...  thus  form  an  orthogonal  system  of  the  type 
considered  by  P.  J.  Daniell*.  This  definition  of  orthogonal  potentials  is 
easily  extended  by  using  a  type  of  integral  suggested  by  the  appropriate 
problem  in  the  Calculus  of  Variations. 

For  the  unit  circle  itself  the  orthogonal  polynomials  are 

Pn(*)  =  &.(n+  1)*, 
and  the  mapping  function  is  consequently  given  by  the  equations 


f(z\  =  v1  -  au)  vz  -  <*) 
§  4-93.   Fejer's  theorem.   Let 

be  the  function  mapping  a  region  D  in  the  Z-plane  on  the  unit  circle  d  with 

*  Phil.  Mag.  (7),  vol.  ii,  p.  247  (1926). 


Fejer's  Theorem  327 

equation  |  z  \  <  1  in  the  z-plane.  We  shall  suppose  that  D  is  bounded  by  a 
Jordan  curve  C  and  that  Z  =  H  (9)  is  the  point  on  G  which  corresponds  to 
the  point  z  =  eie  on  the  unit  circle  c  which  bounds  the  region  d.  At  this 
point,  if  the  series  converges 

Z  =  aQ  +  a^19  -f  a2e2ie  +  ...  anein*  +  ... 

=  ^o  +  Wi  +  ^2  +  •••     say.  ......  (2) 

Now  by  Cauchy's  form  of  Taylor's  theorem 


=  0,         n  <  0, 

where  n  is  an  integer  and  the  contour  is  a  simple  one  enclosing  the  origin 
and  lying  within  the  circle  of  convergence  of  the  power  series.  On  account 
of  the  continuity  of  /  (z)  in  d  we  may  deform  the  contour  until  it  becomes 
the  same  as  c  without  altering  the  values  of  the  integrals.  Hence,  writing 

£  =  em  we  get 

f2jr  foo 

2?7  an  =        H  (a)  e~ina        n  >  0,         0  =      //  (a)  einada. 
Jo  Jo 

These  equations  show  that  the  series  (2)  is  the  Fourier  series  of  the 
continuous  function  H  (6)  and  is  consequently  summable  (<7,  1)  (§  1*16). 

Now  consider  a  circle  j  z  \  =  p  where  p<  1.  The  function  /  (z)  maps  the 
interior  of  this  circle  on  the  interior  of  a  region  R  whose  area  A  is,  by 
§  4-91,  equal  to  the  convergent  series 

7r[|a1|V+2|a2|V+-"]- 
This  area  A  is  bounded  for  all  values  of  p  and  is  less  than  B,  say, 

•'•  "[KIV  +  2  I  «2lV  +  ...n\an\*p*n]<B 
for  0  <  p  <  1  and  so 

7r[|a1|a+2|o2|2  +  ...n\an\*]<B. 

This  inequality  shows  that  the  series  S  n  \  an  |2  is  convergent.  Now  this 
property  combined  with  the  fact  thfrt  (2)  is  summable  ((7,  1)  is  sufficient 
to  show  that  the  series  (2)  converges.  Writing 

sn  -  UQ  +  u^  +  ...  un,     (n  +  1)  Sn  =  50  +  $!  +  ...  sn 

we  have  Sn  =  sn  -  an  where  (n  +  1)  crn  =  t^  +  2^  +  ...  nun.  It  is  suffi- 
cient then  to  show  that  an  ->  0  as  /i  ->  oo.  With  the  notation  #n  =  |  i^n  | 
we  have  the  inequality 

[(m  f-  1)  vm+1  -f  ...  (m  +  p)  vm  +  P]2 

<  [(m  +  1)  +  ...  (m  +  p)]  [(m  +  1)  v2m+1  +  ...  (m  +  p)  v*m+J>]. 

The  first  factor  on  the  right  is  less  than  1  -f  2  4-  ...  (m  +  p)  which  is  less 
than  (m  +  p  +  I)2.  Also,  since  the  series  2rwn2  converges  we  can  choose 


328  Conformal  Representation 

m  so  large  that  the  second  factor  is  less  than  e2  whatever  p  may  be.  We 
may  now  write  n  =  m  +  p, 

*  =  v\  +  202  +  •••  mvm     (w  +  1)  vm+l  +  ...  (m  +  p)  vm+J> 
°n    ~~        m  -}_  p  -j-  i  m  _+_  p  -f  1  > 

where  the  second  term  on  the  right  is  less  than  e  and  since  p  is  at  our 
disposal  we  may  choose  it  so  large  that  the  first  term  on  the  right  is  less 
than  e.  Hence  we  can  choose  n  so  large  that  |  vn  \  <  an*  <  2e  and  so 
|  an  |  ->  0  as  n  ->  oo. 

It  follows  then  that  the  series  (2)  converges  and  that  the  co-ordinates 
(X,  Y)  of  a  point  on  a  simple  closed  Jordan  curve  can  be  expressed  as 
Fourier  series  with  6  as  parameter.  The  theorem  implies  that  the  series  (1) 
converges  uniformly  throughout  d  and  that  the  mapping  by  means  of  the 
function  /  (z)  may  be  extended  to  regions  which  are  slightly  larger  than 
d  and  D. 

Reference  is  made  to  Fejer's  papers,  Milnchener  Sitzungsber.  (1910), 
Comptes  Rendus,  t.  CLVI,  p.  46  (1913)  for  further  developments.  Also  to  the 
book  by  P.  Montel  and  J.  Barbotte,  Lemons  sur  les  families  normales  de 
fonctions  analytiques  (Gauthier-Villars,  Paris,  1927),  p.  118. 


CHAPTER  V 
EQUATIONS  IN  THREE  VARIABLES 

§  5-11.  Simple  solutions  and  their  generalisation.  Commencing  as  before 
with  some  applications  of  the  simple  solutions  we  consider  the  equation 


_ 

p  W  ~  ^  \d&  + 

of  the  propagation  of  Love-waves  in  the  direction  of  the  #-axis.    If  now 
p  and  fjb  have  the  values  p0  ,  /ZQ  respectively  for  z  >  0,  and  the  values  p±  ,  p^ 
respectively  for  z  <  0,  it  IP  useful  to  consider  a  solution  of  type 
v  =  v0  =  (A  cos  sz  -f  B  sin  sz)  sin  K  (x  —  qt)3-     z  >  0, 

v  =  Vl  =  Cfcto  sin  K  (x  —  gtf),     2  <  0, 
where  the  constants  are  connected  by  the  relations 

A»*V  =  Mo  (*2  +  *2)>     Pi^V  =  Mi  (^2  ~  *a)- 

In  the  expression  for  v2  we  take  h  >  0  so  that  there  is  no  deep  pene- 
tration of  the  waves. 

The  boundary  conditions  are 

/^  -~  =  0,  when  z  =  a, 


These  equations  give 

4  sin  sa  =  J5  cos  sa, 

4  =  (7, 


Putting  ^  =  C02p0  ,  ^  =  q^!  we  have  with  s  =  K  cosech  o>0  ,  A  =  K  sech  o^  , 
c0  =  g  tanh  o;0  ,     cx  =  y  coth  cox  , 

cosh  Wi  =  —  sinh  o>0  cot  (CLK  cosech  co0)  =  (  1  —  ^  tanh2  o>0  )    , 
Mo  V        ci  ' 

and  it  is  readily  seen  that  there  are  no  waves  of  the  present  type  unless 
c0<c1. 

Matuzawa  has  examined  the  case  of  three  media  arranged  so  that  in 
his  notation 

v  =  vl  =  Al  (e°iz  +  e-*iz)  cos  (pt  +  fx),      p  =  p1?     M  =  Mi>     0  >  2  >  -  A, 
v  =  vz  =  (^2e**z  +  Be~'*z)  cos  (jp£  -f  /»),    p  =  />2J     M==M2>     —  h>  z>  —  H, 

V  =  V3  =  ^36*8*  COS  (p^  -h  /#),  P  =  P3>       M^MSJ       ""  H  >  Z. 


330  Equations  in  Three  Variables 

The  boundary  conditions 

dvl  dvz     .  , 

^i  -  v2,     to  -      =  ft  ~      at  z  -  - 


give  ^4j  (e~*iA  + 


Eliminating  ^ij,  ^42»  ^3>  ^2  and  writing  rx  =  tanh  sji,  r2  =  tanh  s2h, 
T2  =  tanh  s2H,  we  obtain  the  equation 


?  (T2  -    T,) 


The  cases  ^,  52,  Ǥ3  all  real  and  52  imaginary,  sl9  s3  real,  are  not  com- 
patible with  an  equation  of  this  type.  When  s2  is  real  it  appears  that  there 
is  only  one  value  of  sl  and  this  is  an  imaginary  quantity;  when  .$2  is  an 
imaginary  quantity  it  appears  that  there  are  two  possible  values  of  sl  and 
these  are  both  imaginary. 

Matuzawa  has  examined  the  six  possible  cases 
A  B  G  D  E  F 

Cl<C2<  C3      Cl<C3<  C2      C2<Cl<  C3      C3  <  G!  <  C2      C2<C3<  Ct      C3  <  C2  <  Cx 

and  concludes  that  in  cases  B,  D  and  F  there  is  no  solution. 

§  5*12.  The  simple  solutions  considered  so  far  correspond  to  the  case 
of  travelling  waves.  We  shall  next  consider  a  case  of  standing  waves  and 
shall  take  the  equation  of  a  vibrating  membrane 


dt2  ~~      \dx* 

Let  the  boundary  of  the  membrane  consist  of  the  axes  of  co-ordinates 
and  the  lines  x  =•  a,  y  =  b.  The  expression 

.     m-rrX    .     UTTV  ,  4  ^         .        .. 

w  =  sin  —  —  sin  -^  {^4mn  cos  pt  +  Bmn  sin  jrf} 
w  c/ 

satisfies  the  condition  w  =  0  on  the  boundary  and  is  a  simple  solution  of 
the  differential  equation  if 


This  equation  gives  the  possible  frequencies  of  vibration,  ra  and  n 
being  integers.  A  more  general  type  of  vibration  may  be  obtained  by 
summing  with  respect  to  m  and  n  from  m  =  1  to  oo  and  n  =  1  to  oo. 


Standing  Waves  331 

The  resulting  double  Fourier  series  is  usually  a  solution  which  is  sufficiently 
general  to  make  it  possible  to  satisfy  assigned  initial  conditions 

w  ^  WQ'      dt  =  ^°  f°r  ^==0> 

by  using  coefficients  Amn,  Bmn  determined  by  Fourier's  rule 
4  4   [a  f&         .    mrrx    . 


a  6  nny 

- 


4    fa  f6  .          mTTX    . 
n  =  ~T—          wo  sin  s 

a6jpJoJo  a 


EXAMPLE 

Find  the  nodal  lines  of  the  solutions 


.     .     TTX    .     *rrtj  f         nX  7rtJ\ 

>  =  A  sm  -  -  sm  --  (cos h  cos  —  )  cos  pt, 

a         a  \       a  a  /        ^ 


A     .  '27TX    .     2rry 

w  =  A  sin  —  sin  —  -  cos  pt, 
a  a 

~  (  .     SKX    .Try         .     TTX    .     STT^I 
w  —  C  {sin  —  sin     -  —  sin  —  sin  —  ->  coapt, 

\        a          a  a          a  j  • 

which  are  suitable  for  the  representation  of  the  vibration  of  a  square  if  p  has  an  appropriate 
value  in  each  case. 

§  5-13.    Reflection  and  refraction  of  electromagnetic  waves.    In  a  non- 
conducting medium  the  equations  of  the  electromagnetic  field  are* 
curl  H  -  J9/c,       div  D  =  0, 
curl  E  =  -  B/c,   div  B  =  0, 
and  the  constitutive  relations  are 

D  -  xE,    B=  /*//, 

where  the  coefficients  K  and  ^  can  be  regarded  as  constants  if  the  material 
is  homogeneous  and  the  frequency  of  the  waves  is  not  too  high. 

If  all  the  field  vectors  are  independent  of  z,  their  components  satisfy 
the  two-dimensional  wave-equation 


where  V2  =  c2/Kp,  =  1/s2,  say. 

The  permeability  of  all  substances  is  practically  unity  for  frequencies  as 
great  as  that  of  light.  Hence  for  light  waves  it  is  permissible  to  write 

V  =  C/VK, 

and  in  this  case  we  may  also  write  /c  =  n2,  where  n  is  the  index  of  refraction 
of  the  medium. 

Let  us  now  suppose  that  the  medium  with  the  constants  (K^,  /^)  is  on 

*  For  convenience  we  denote  a  partial  differentiation  with  respect  to  the  time  t  by  a  dot. 


332  Equations  in  Three  Variables 

the  side  x  <  0  of  the  plane  x  =  0,  and  that  on  the  other  side  of  this  plane 
there  is  a  medium  with  constants  (/c2,  /x2). 

We  shall  suppose  that  when  x  <  0  there  is  an  incident  and  a  reflected 
wave,  but  that  for  x  >  0  there  is  only  a  transmitted  wave.  We  shall 
suppose  further  that  the  electric  vector  in  all  the  waves  is  parallel  to  the 
axis  of  2,  then  with  a  view  of  being  able  to  satisfy  the  boundary  conditions 
we  assume 

Ez  -  A&  +  AM   (x  <  0),     E,  =  A2e2   (x  >  0), 

where  e^,  e±  ',  e2  denote  respectively  the  exponentials 

g    __  gitofs^j;  co8#i  +  1/  sin  <£>!)-']        g  ' 


The  corresponding  expressions  for  the  components  of  H  are 
Hx  -  (csjfr)  (A&  4-  AM)  sin  </>!>     #  <  0, 
Hx  =  (c52//LL2)  ^4262  sin  </>2>  a;  >  0, 

//„  =  (c$i/  /L4)  (A'e/  ~  ^iei)  cos  </>!,    a  <  0, 
Hy  =  —  (cs2/iJL2)  A2e2  cos  </>2,  a;  >  0. 

The  boundary  conditions  are  that  the  tangential  components  of  E  and 
//  are  to  be  continuous.  These  conditions  give 

A.}  -f-  AI   =  A2, 
sin  fa.ptSi  (Al  -f  AI)  =  /^i«§2^2  sin  ^2» 

COS  <>i.X2<Sl  (-^1  —  ^/)  =   ^152^2  COS  <2- 


TT  l  l2  2  l 

Hence  .    ^  =  ^-  =  —     when  ux  =  u2. 

sin  (f>2      p,2si      ni 


This  is  the  familiar  relation  of  Snell.  Writing 

A9  =  fi-4,     ^2  =  T^, 

where  7?  and  T  are  the  coefficients  of  reflection  and  transmission  re- 
spectively, we  have 

1  —  R  _  sin  ^  cos  fa  T*  _       gin  (^i  —  ^2) 

1  -}-  jR      sin  <^2  cos  </>!  '  sin  (0t  +  ^2)  ' 

2  —  T      sin  <^x  cos  ^>2  ^      2  sin  ^>2  cos  ^ 
27      ~"  sin  ^2  cos  </>!  '         ~~    sin  (fa  4-  <^2) 

In  the  case  when  the  electric  vector  is  in  the  plane  of  incidence  we  write 
EX  -  -  (C&  +  CM)  sin  <£i>  EX  =  -  ^2^2  sin  </>2, 

^  =  ((7^  -  CW)  cos  ^    (a;  <  0),        Ev  --=  ^63  cos  <£2   (x  >  0), 
H,  =  (c^/^)  (OlCl  +  CM),  Hz 

and  the  boundary  conditions  give 


—  GI)  cos  ^  =  (72  cos  ^2 
-f  Cf1/)  sin  </>!  =  ^^  sin 


Reflection  and  Refraction  333 

The  third  equation  implies  that  the  ^-component  of  D  is  continuous 
at  x  =  0.  These  equations  give 
sin  <A1 


= 
Sin 


Thus  Snell's  law  holds  as  before.   If  we  further  write 

CS-pCv     Ct=rCt, 
so  that  p,  T  are  the  coefficients  of  reflection  and  transmission  respectively, 

WC  haVe  =  tan  (A  -  c£2) 

p      tan  (^  +  </>2)  ' 

2  sin  <£2  cos  (/>! 
r  ~  sin  (<£j,  +  <f>2)  cos  (</>!  -  ^>2)  ' 

Of  the  four  quantities  R,  T,  py  r  only  one  can  vanish,  viz.  the  polarizing 
angle  Ox  is  defined  as  the  angle  of  incidence  for  which  p  =  0.  This  angle 
is  given  by  the  equation  tan  (^  -f-  <f>2)  =  oo,  and  so 

tan  Oj  —  n2/n1  . 

When  the  incident  light  is  unpolarized  it  consists  of  a  mixture  of  waves 
in  some  of  which  E  is  parallel  to  the  axis  of  z  and  in  the  others  H  is  parallel 
to  the  axis  of  z.  When  such  light  strikes  the  surface  x  =0  at  the  polarizing 
angle  the  waves  of  the  second  kind  are  transmitted  in  toto,  and  so  the 
reflected  light  consists  merely  of  waves  of  the  first  kind  and  is  thus  linearly 
polarized. 

Reflection  and  refraction  of  plane  waves  of  sound.  Consider  a  homo- 
geneous medium  whose  natural  density  is  p0.  When  waves  of  sound 
traverse  the  medium  the  density  p  and  pressure  p  at  an  arbitrary  point 
Q  (%>  y,  z)  have  at  time  t  new  values  which  may  be  expressed  in  the  forms 

p  =  Po(l  +  5),    p  =  p0(l  4-  As), 

where  p0  is  the  undisturbed  pressure  and  A  is  a  coefficient  depending  on 
the  compressibility.  The  quantity  s  is  called  the  condensation  and  will  be 
assumed  to  be  so  small  that  its  square  may  be  neglected. 

We  now  suppose  that  the  velocity  components  (u,  v,  w)  of  the  medium 
at  the  point  Q  can  be  derived  from  a  velocity  potential  <f>  which  depends 
on  the  time.  Bernoulli's  integral 

[dp      d<f>  ,      . 

\  -   +  -7T7  =  constant 
]  P        dt 

then  gives  the  approximate  equation 


where  c2  =  pQA/pQ  =  (dpjdp)Q  is  the  local  velocity  of  sound  and  is  constant 
since  A  and  pQ  are  constants.  The  equation  of  continuity 


334  Equations  in  Three  Variables 

and  the  equations  u  ~  ~  ,  v  =  -~  ,  w  =  ^,  when  ?/,  t;,  w  are  small,  give 


the  wave-equation 


<>i 


The  conditions  to  be  satisfied  at  the  surface  separating  two  media  are 
that  the  pressure  and  the  normal  component  of  velocity  must  be  con- 

tinuous.   On  account  of  Bernoulli's  equation  the  continuity  of  pressure 

p  / 

implies  that  p  ^~  is  continuous. 

Let  us  now  consider  the  case  in  which  two  media  are  separated  by  the 
plane  x  —  0.  We  shall  suppose  that  in  the  medium  on  the  left  there  is  an 
initial  train  of  plane  waves  represented  by  the  velocity  potential 

^  =  aoC«»«-f»-«>, 

and  that  these  waves  are  partly  reflected  and  partly  transmitted.    We 
therefore  assume  that 

(/>!  =  a0etn(t-tx-™}  +  a1ein(e+ffl?-1iy),     x  <  0, 
</>2  =  a2etn(t-Sx-™}  ,  x  >  0. 

The  boundary  conditions  give 

Pi  (0'0+  %) 


where  pl  and  p2  are  the  values  of  the  natural  density  for  x  <  0  and  x  >  0 
respectively.  If  cx  and  c2  are  the  two  associated  velocities  of  sound,  we 
must  have  c^cos^,  c^sin^, 

c2  £  =  cos  «2  >     C27?  ^  sin  a2  • 
Therefore 


__              cos  «!  —  clpl  gos  «2  ^2  c°t  ai  ~"  Pi 

1            °              COS  al  +  ClPl  COS  ^2  °  P2  CO^  al  +  Pl 

cos  %            _  2/>!  cot 

"                    -     "  ~                     " 


—  —         ri  .  . 

COS  «!  -h  C1pl  COS  a2  /02  COt  «!  -f-  /)j  COt  a2 

The  equation  c2  sin  %  =  ct  sin  «2 

gives  a  law  of  refraction  analogous  to  Snell's  law. 

When  the  second  medium  ends  at  x  =  6,  where  b  >  0,  and  for  x  >  b 
the  medium  is  the  same  as  the  first,  there  are  three  forms  for  the  velocity 
potential:  ^  ^  ^ginu-f*-™)  +  a^in(t^x-^y^  x  <  0, 

</>2  =  a2ein(t"^-^  +  a3einU+^-^>,     6  >  a;  >  0, 
(/>3  -  a4emU-^-^>, 
and  the  boundary  conditions  give 

pi  K  -f  «i)  ==  p2  K  +  «3)»    ^  K  -  «i)  =  £  («2  -  %)> 


Energy  Equation  335 

Therefore 

cos  (*6£)  +  -^  sin 

4 

;  sin  (n&£)  4-  ^ 
4 

=  «       os  (n£)  +  Jt  i2  +          sn 
£-2  -  M  sin 

bPl          £P2J 

It  should  be  noticed  that  these  equations  give 

Kl2-  I  oil2-  Kl2, 

I  19  I  19  I  19 

p2  1  «2  12  -  />2  1  %  r  =  PI  I  ^  r  cot  a  , 

and  the  first  of  these  equations  indicates  that  the  sum  of  the  energies  per 
wave-length  of  the  reflected  and  transmitted  waves  in  the  first  medium  is 
equal  to  the  energy  per  wave-length  in  the  incident  wave. 

It  should  be  noticed  that  if  sin  (nb£)  ^  0,  the  condition  for  no  reflected 
wave  (%  =  0)  is  £p2~  £pi>  and  is  independent  of  the  thickness  of  the 
second  medium. 

We  have  assumed  so  far  that  there  is  a  real  angle  «2  which  satisfies  the 
equation  c±  sin  0%  —  c2  sin  «x  ,  but  if  c2  >  Cj  it  may  happen  that  there  is  no 
such  angle.  If  the  value  of  sin  «2  given  by  this  equation  is  greater  than 
unity,  cos  «2  will  be  imaginary  and  the  solution  appropriate  for  a  single 
surface  of  separation  (x  —  0)  will  be  of  type 

</>x  -  a0ein(t^x-^  -f  a1etnU+^-Tjy),     x  <  0, 

^2  =  o2et"n(*-1»')-*a!,  x  >  0,  0  >  0. 

In  this  case  there  is  no  proper  wave  in  the  second  medium,  and  on 
account  of  the  exponential  factor  e~6x  the  intensity  of  the  disturbance  falls 
off  very  rapidly  as  x  increases.  The  corresponding  solution  of  the  problem 
for  the  case  in  which  the  second  medium  is  of  thickness  b  is  obtained  from 
the  formulae  already  given  by  replacing  £  by  —  id.  It  is  thus  found  that 


P* 


-- 
LP2        0J 

a4  [cosh  (nbO)  4-  \i  (^  -  ^}  sinh  (nb0)  I  , 

L  Wl          Sp2/  J 

2  -f  ^l  sinh  (nbO). 


The  coefficient  a3  of  the  disturbance  of  type  einu"liv)+a*  which  increases 
in  intensity  with  a;  is  seen  to  be  very  small  so  that  this  disturbance  is  small 
even  when  x  =  6. 


336  Equations  in  Three  Variables 

In  the  present  case  the  reflection  is  not  quite  total,  for  some  sound 
reaches  the  medium  x  >  b.  The  change  of  phase  on  reflection  is  easily 
calculated  by  expressing  aj/a0  in  the  form  jRe*". 

Let  us  now  consider  briefly  the  case  when  nb£  —  krr,  where  k  is  an 
integer.  In  this  case  sin  (nb£)  =  0;  there  b  no  reflected  wave  and  the 
formulae  become  simply 

</»i  =  a^ein(t~^-^\  x  <  0, 

</>2  •=  «o  Wp2)  ein(t-*x-™>  +  a3ein(t~™>  sin  (£nx),    b  >  x  >  0, 
(f>3  =  a0etnU~f*-'I1'),  x  >  b. 

It  will  be  noticed  that  the  value  of  n  is  precisely  one  for  which  there  is 

a  potential  </>  fulfilling  the  conditions  -~  =  0  for  x  =  0  and  x  =  b. 

The  slab  of  material  between  #  =  0  and  x  =  b  can  be  regarded  as  in  a 
state  of  free  vibration  of  such  an  intensity  that  there  is  no  interference 
with  the  travelling  waves. 

The  absorption  of  plane  waves  of  sound  by  a  slab  of  soft  material  has 
been  treated  by  Rayleigh*  by  an  ingenious  approximate  method  in  which 
the  material  is  regarded  as  perforated  by  a  large  number  of  cylindrical 
holes  with  axes  parallel  to  the  axis  of  x  and  the  velocity  potential  within 
these  holes  is  supposed  to  satisfy  an  equation  of  type 


where  h  is  a  positive  constant.  The  new  term  is  supposed  to  take  into 
consideration  the  effect  of  dissipation. 

At  a  very  short  distance  from  the  mouth  (x  =  0)  of  a  channel  it  is 

£}2JL  PV2JL 

assumed  that  the  terms   ~  -2  and  ^  ^   may  be  neglected  and  that  the 
solution  is  effectively  of  type 

£  =  eint  {a!  cos  k'x  -4-  6'  sin  k'x}, 
where  c2k'2  =  n2  —  inh. 

If  the  channel  is  closed  at  x  =  6,  we  have  ^  -  =  0  there,  and  so  we  may 

write 

</»  =  ^4Vn'cosfc'  (x—  b). 

When  x  is  very  small 


C25  =  _  _r  =  _  inA'eini  cos  (k'b), 
u       ik' 


c*s       n         {k'b}- 

*  Phil.  Mag.  (6),  vol.  xxxix,  p.  225  (1920);  Papers,  vol.  vi,  p.  662. 


Reflection  337 

If,  for  x  <  0,  we  adopt  the  same  expression  as  before,  viz. 


we  have 

Now  let  a  be  the  perforated  area  of  the  slab  and  v'  the  area  free  from 
holes.  The  transition  from  one  state  of  motion  on  the  side  x  <  0  to  the 
other  state  on  the  side  x  >  0  is  assumed  to  be  of  such  a  nature  that 

(cr  -f  a')  ul=  cm, 


These  equations  give  the  relation 
a0  —  a*       ikf 


.-.  ,,  . 
,  tan  (kb) 


a0      ai      n$  <* 

for  the  determination  of  the  intensity  of  the  reflected  wave.  When  h  —  0, 
we  have  |  ax  |  =  |  a0  |  and  the  reflection  is  total,  as  it  should  be.  When 
a  =  0,  a±  =  a0,  and  there  is  again  total  reflection.  On  the  other  hand,  if 
a  =  0,  the  partitions  between  the  channels  being  infinitely  thin,  we  have, 
when  h  =  0,  * 

_      ng  cos  (k'b)  —  ik'  sin  (k'b)  __      cos  «j  cos  (k'b)  —  i  sin  (k'b) 
1  ~    °  ?i£  cos  (&'6)  -f   ^'  sin  (k'b)  ~    °  cos  «j  cos^F6)  4-  *  sin  (k'b)  ' 


In  the  case  of  normal  incidence  «x  =  0,  ax  =  a0e~2ifc/b,  and  the  effect  is 
the  same  as  if  the  wall  were  transferred  to  x  =  b.  When  h  is  very  small 
but  the  term  k2  in  the  complex  expression  k'  =  k^  +  ik2  is  so  large  that 
the  vibrations  in  the  channels  are  sensibly  extinguished  before  the  stopped 
end  is  reached,  we  may  write 

cos  (ik2b)  =  Jefc2&,     sin  (ik2b)  —  \iek^,     tan  (k'b)  =  —  i, 
and  the  formula  becomes 


a0  +  di       (cr  +  a')  cos  a±' 

EXAMPLES 

1.  In  the  reflection  of  plane  waves  of  sound  at  a  plane  interface  between  two  media 
the  velocity  of  the  trace  of  a  wave-front  on  the  plane  interface  is  the  same  in  the  two  media. 

*  [Rayleigh.] 

2.  When  the  velocity  of  sound  at  altitude  z  is  c  and  the  wind  velocity  has  components 
(u,  v,  0),  the  axis  of  z  being  vertical,  the  laws  of  refraction  are  expressed  by  the  equations 

<f>  =  <t>Q ,     c  cosec  9  -h  u  cos  <f>  +  v  sin  <f>  =--  c0  cosec  00  +  UQ  cos  <£0  +  v0  sin  <£0  =  A,  say, 

where  (0,  <f>)  are  the  spherical  polar  co-ordinates  of  the  wave-normal  relative  to  the  vertical 
polar  axis  and  the  suffix  0  is  used  to  indicate  values  of  quantities  at  the  level  of  the  ground. 

3.  Prove  that  the  ray- velocity  (the  rays  being  defined  as  the  bicharacteristics  as  in 
§  1*93)  is  obtained  by  compounding  the  wind  velocity  with  a  velocitj7  c  directed  along  the 
wave-normal.   See  also  Ex.  1,  §  12-1. 


338  Equations  in  Three  Variables 

4.  The  range  and  time  of  passage  of  sound  which  travels  up  into  the  air  and  down  again 
are  given  by  the  equations 

rz 

x  =  2       (c2  cos  <£  -f  us)  dz/T, 
J  o 


fZ 

y  =  21     (c2  sin  <f>  +  vs)  dz/r, 
J  o 


fZ 

*  =  2      sdz/r, 
Jo 

where  s  —  A  —  u  cos  <f>  ~  v  sin  <f>,    r  —  s  (s2  —c2)*, 

and  Z  is  defined  by  the  equation  s  =  c. 

§  5-21.  Some  problems  in  the  conduction  of  heat.  Our  first  problem  is  to 
find  a  solution  of  the  equation 


which  will  satisfy  the  conditions 

6  —  exp  [ip  (t  —  x/c)]  when  y  =  0,     6=0  when  y  =  oo. 
Assuming  as  a  trial  solution 

exp  [ip  (t  -  &/c  -  y/b)  -  ay], 

Kip\  2      7?2~l 
a  +  ~-j   —  ^-g    • 

Therefore  6  =  2a/c,    p2  (  ~  -f  ~)  =  a2. 

-r    ^2       cay 

The  result  tells  us  that  if  the  temperature  at  the  ground  (y  =  0)  varies 
in  a  manner  corresponding  to  a  travelling  periodic  disturbance,  the  variation 
of  temperature  at  depth  y  will  also  correspond  to  a  periodic  disturbance 
travelling  with  the  same  velocity  but  this  disturbance  lags  behind  the 
other  in  phase  and  has  a  smaller  amplitude. 

The  solution  may  be  generalised  by  writing 

b  =  c  tan  </>,     a  =  (c/2/c)  tan  <£,    p  =  (c2/2/c)  tan  (/>  sin  0, 

7T 

f2 

#  =      f  (<f>)  d<f>.  exp  [(ic/2/c)  (c£  —  x)  tan  <f>  sin  <^  —  (cy/2/c)  (tan  (/>  +  i  sin  ^)], 
Jo 

where  c  i^>  regarded  as  a  constant  independent  of  (f>  and  /  (c/>)  is  a  suitable 
arbitrary  function. 

If  we  wish  this  solution  to  satisfy  the  conditions 

0  =  g  (ct  —  x)  when  y  =  0,     0  =  0  when  y  =  oo, 
the  function  /  (<£)  must  be  derived  from  the  integral  equation 


f 

= 

Jo 


2*)  tan  <f>  sin  </>],     (—  oo  <  u  <  oo). 


Solution  'by  Definite  Integrals  339 

When  the  function  g  (u)  is  of  a  suitable  type,  Fourier's  inversion 
formula  gives 

foo 

/  (<£)  ^  (c/47T/c)        (1  4-  sec2  </>)  sin  <f>.g  (u)  du.exp  [—  (icu/2f<)  tan  </>  sin  </>], 

J-oo 

0  <    </>  <   7T/2. 

In  particular,  if 

</  (tf)  =  (2K/cu)  sin  [tan  a  sin  a  (c^/2/c)], 
where  a  is  a  constant,  we  have 

6  =  f  "sin  <f>  (1  +  sec2  0)  dJ>  .  e~  <<*/2«>tan* 
Jo 

x  sin  [(c/2/c)  {(cZ  —  x)  tan  </>  sin  </>  —  y  sin  </>}]. 

Another  solution  may  be  obtained  by  making  c  a  function  of  <f>  and  then 
integrating  ;  for  instance,  if  c  =  2/c  cos  (/>  we  obtain  the  solution 

TT 

f2 

0  =      f  ((f>)  d<f>.  exp  [i  sin2  <f>  (2i<t  cos  (^  —  x)  —  y  (sin  ^  -f  i  sin  <f>  cos  <^)]. 
Jo 

It  should  be  noticed  that  the  definite  integral 

7T 

f2 

0  (#,  y,z,t)=     f((f))d</).  exp  [i  sin2  0  (2/c^  cos  6  —  x)  —  z  sin  </>  —  iy  sin  <b  cos  <i] 

Jo 

is  a  solution  of  the  two  partial  differential  equations 


and  is  of  such  a  nature  that  the  function 

9  (x,  y,  t)  =  0  (x,  y,  y,  *) 

is  a  solution  of  equation  (A).  It  is  easy  to  verify,  in  fact,  that  if  0  (x,  y,  z,  t) 
is  any  solution  of  equations  (B)  the  associated  function  0  (x,  y,  I)  is  a 
solution  of  (A),  for  we  have 

d*e    a^^a2©    a2©    a2©       a2© 

3z2  +  a*/2  ~  a*2  +  dy*  +  dz2          dydz 

=  2    —  =  1—  =1  8^ 
dydz      K  dt      K  dt  ' 

Again,  if  we  take  c  =  2/c  cot  <f>,  we  obtain  an  integral 

IT 

(2 

0  (x,  y,  t)  =     /  (</>)  d<f>.exp  [i  (2.Kt  cos  (f>  —  x  sin  </>)  —  y  (I  -f  i  cos  <^>)], 
Jo 

which  is  a  solution  of  (A),  and  the  associated  integral 

7T 

(2 

0  (x,  y,  z,t)=\f  (<f*)  d<f>  .  exp  [i  (2/c£  cos  <j>  —  x  sin  <f>  —  y  cos  <£)  ^-  2] 
Jo 

......  (C) 


340  Equations  in  Three  Variables 

is  likewise  a  solution  of  the  equations  (B).  Indeed,  if  c  is  any  suitable 
function  of  cf>  the  integral 


r 

= 
J 


(ct  —  #)  tan  </>  sin  </> 
—  (c/2/c)  (2  tan  (f>  +  iy  sin  </>)] 


is  a  solution  of  the  equations  (B). 

It  should  be  noticed  that  the  particular  solution  (C)  is  of  type 

0  (x,  y,  z,  t)  =  e~*F  (x,  y  -  2Kt), 
where  F  (u,  v)  is  a  solution  of  the  equation 


This  indicates  that  if  F  is  any  solution  of  this  equation,  then  the 
function  9  ^  y^  t)  =  \_vp  {Xt  y  _  2Kt) 

is  a  solution  of  the  equation  (A).  This  is  easily  verified  by  differentiation. 
Since  there  is  also  a  solution  0  =  erKtF  (x,y)9  we  have  two  different  wrays 
of  deriving  a  particular  solution  of  the  equation  (A)  from  a  particular 
solution  of  the  equation  (D). 

Since  F  (u,  v)  =  J0  \/\u*  -f  v2]  is  a  particular  solution  of  equation  (D) 
there  is  a  certain  surface  distribution  of  temperature 

6  ^  J0  V|z2  -f  4fc2*2],     when  y  =  0, 

which  is  propagated  downwards  as  a  travelling  disturbance  gradually 
damped  on  the  way,  the  velocity  of  propagation  being  2/c. 

If,  on  the  other  hand,  we  take  F  (u,  v)  =  cos  mu.exp  v  [m2  —  l]i,  we 
obtain  a  distribution  of  temperature 

0  (x,  y,  t)  =  e-»  cos  mx.exp  {(y  -  2*0  [m2  -  1]!},     m2  >  1     ......  (E) 

in  which  a  periodic  surface  distribution  is  decaying  at  the  same  proportional 
rate  at  every  point  of  the  surface.  If  m2  <  2  the  foregoing  distribution 
gives  0  =  0  when  y  =  oo.  The  periodic  distribution  now  travels  upwards 
with  constant  velocity 

c  -  2*  (m2  -  !)*/[!  -  (m2  -  1)*], 

and  the  rate  of  damping  at  depth  y  is  the  same  as  that  at  the  surface,  but 
at  any  instant  the  temperature  at  this  depth  is  a  fraction 

exp[-  1  +  (m2-  1)4] 
of  that  at  the  surface.  When  m2  =  2  there  is  a  distribution  of  temperature 

9  -  e-2*<  cos  (x  -v/2), 

which  is  independent  of  the  depth  but  does  not  satisfy  the  condition* 
0=0  when  y  =  oo.  When  m2  >  2  the  distribution  (E)  gives  6=0  when 

*  In  this  case  there  is  no  solution  of  type  0  =  e~2ltt  Y  (y)  cos  (x  N;2)  which  gives  the  foregoing 
surface  value  of  t  and  a  value  0=0  when  y  =  oo  for  Y"  (y)  —  0. 


Circular  Source  of  Heat  341 

y  =  —  oo,  and  the  material  into  which  conduction  takes  place  may  be 
supposed  to  be  on  the  side  y  <  0.  In  this  case  the  velocity  of  propagation  is 

c  =  2*  (m2  -  l)4/[(ra2  -  1)*  -  1], 
and  the  temperature  at  depth  |  y  \  is  at  any  instant  a  fraction 

exp  -  [(m2  -  1)1  -  1] 
of  that  at  the  surface. 

We  have  seen  in  §  2*432  that  if  0  (x,  y,  t)  is  a  solution  of  equation  (A) 
then  the  function 


9  -, 

is  a  second  solution.   If,  in  particular,  we  take  the  function 

6  (x,  y,  t)  =  er«  F  (x,  y), 
where  F  (u,  v)  satisfies  (D),  we  obtain  the  solution 


If  r2  =  x2  H-  i/2  there  is  a  solution 


</>  =  ^e       4^     J0(arlt)  ......  (G) 

depending  only  on  r  and  t  which  at  time  t  —  0  is  zero  at  all  points  outside 
the  circle  r  =  2a/c.  When  t  >  0  the  temperature  at  points  of  the  circle  is 
given  by  </>  =  t~lJ0  (2a2K/t).  The  circle  can  thus  be  regarded  as  a  source  of 
fluctuations  in  temperature  which  are  transmitted  by  conduction  to  the 
external  space.  The  total  flow  of  heat  from  this  circular  source  in  the 
interval  t  =  0  to  t  =  oo  may  be  obtained  by  calculating  the  integral 


dt. 


Now  =     -        J0  ( 


Also  [^  dt  (alt2}  e/0'  (2a2K/t)  =  -  l/2*a, 

Jo 

r  °° 

eft  (a/J2)  J0  (2a2*/£)  -       1/2/ca. 
Jo 

Hence  dt  (  ^ }         =  —  l//ca. 

Jo       \^r)r^a< 

and  so  the  total  flow  of  heat  from  the  circle  is 


342  Equations  in  Three  Variables 

This  is  independent  of  a  and  so  our  formula  holds  also  for  a  point 
source.  The  temperature  function  of  a  point  source  of  "  strength"  Q  is  thus 

^  =  (ty*™*)-1^/4*  ......  (H) 

while  that  of  a  circular  source  is 


)'ie  J0  (ar/0-  ......  (I) 

This  result  is  easily  extended  to  a  space  of  n  dimensions,  thus  in  three- 
dimensional  space  the  temperature  function  for  a  spherical  source  of 
strength  Q  is 


t)-*e         *     sm(ar/t)/(ar/t).  ......  (J) 

The  solution  for  an  instantaneous  source  uniformly  distributed  over 
a  circular  cylinder  has  been  obtained  by  Lord  Rayleigh*  by  integrating 
the  solution  for  an  instantaneous  line  source.  The  result  is 

r2  +  a2  -  2ar  cos  0  r2  -f  a8 


A  more  general  solution  is 

H  +  a2 


Integration  with  respect  to  t  from  0  to  oo  gives  a  corresponding  solution 
of  Laplace's  equation  and  we  have  the  identity 


dt 

I         I    \  o  -*-""       I    I  ty    ^    ft      \ 

*•  *»    I    /-k       .    I  C*  .      I  I  *     ^--     t€/,   | 

(M) 

r  >  a.  I 


The  temperature  0  due  to  an  instantaneous  line  doubletf  of  strength  q 
may  be  derived  by  differentiating  with  respect  to  y  the  temperature  </> 
due  to  an  instantaneous  line  source  of  strength  q.  Since  the  latter  is 

<f>  -  (g/4^Kt)  e-'2/4"', 
we  have  6  =  (qy^TTKH^)  e~r2lM.  ......  (N) 

The  temperature  due  to  a  continuous  line  doublet  of  constant  strength 
Q  is  obtained  by  integrating  with  respect  to  t  between  0  and  t.  Denoting 
this  temperature  by  0  we  have 

0  =     'flcft  =  (qylZtTKr*)     '  ~  [e-*l*"]  dt  = 


0  =  f'flcft  =  (qylZtTKr*)  f  '  ~  [e-*l*"] 
Jo  Jo  at 


*  Phil.  Mag.  vol.  xxn,  p.  381  (1911);  Papers,  vol.  vi,  p.  51. 

t  See  Carslaw's  Fourier  Series  and  Integrals,  p.  345  (1906).  The  direction  of  the  doublet  is 
that  of  the  axis  of  y.  The  doublet  is  supposed  to  be  "located"  at  the  origin. 


Instantaneous  Doublet  343 

This  solution  may  be  used  to  find  a  solution  of  (A)  which  takes  the 
value  F  (x)  when  y  =  0  and  is  zero  when  y  =  oo  and  when  t  =  0.   If  9  is 

to  be  such  that  9,  ~-  and  -5-  are  continuous  for  y  >  0,  an  appropriate 
expression  for  9  is 

00  rJ      f  (X-X* 

1  yasecaa  \  (0) 

&  __ — _  \     / 

da  F(x-\-  i/tana)e       4l<t    . 


- 

Tt] 


2 

The  first  integral  evidently  satisfies  (A)  if  y  >  0,  and  the  second  integral 
tends  to  F  (x)  as  y  ->  0  if  F  (x)  is  a  continuous  function  of  x. 

In  the  special  case  when  F  (x)  =  1  the  expression  for  9  takes  the  form 

^y*  8eca  a 

""~ 


a 
and  can  be  expressed  in  the  well-known  form* 

277-*      e~v2dv,  (P) 


where  u2  =  y2/4:t<t  and  w  >  0. 

If  the  boundary  y  =  0  is  maintained  at  the  temperature  F  (x,  t)  the 
solution  which  is  zero  when  y  =  oo  and  when  Z  =  0  is  given  by  the  formula 


/  l'^'  dt' 

o  (^  -  t  Y 

There  is  a  similar  formula  for  a  space  of  three  dimensions. 
If  9  =  F  (x,  y,  t)  when  z  =  0  and  0=0  when  z  =  oo  and  when  £  =  0, 
the  appropriate  solution  is 


(B) 

In  this  case  an  element  of  the  integrand  corresponds  to  an  instantaneous 
doublet  whose  direction  is  that  of  the  axis  of  z. 

Let  us  next  consider  a  case  of  steady  heat  conduction  in  a  fluid  moving 
vertically  with  constant  velocity  w.  The  fundamental  equation  is 

'N/J  /  CJ2/J          O2/3 

(7(7  /  O  "         \J   " 

where  K  is  the  diffusivity.    Writing  0  =  ©e"*/2*  the  equation  satisfied  by  0  is 

V20  =  A20, 

*  The  transformation  from  one  integral  to  the  other  can  be  made  by  successive  differen- 
tiation and  integration  with  respect  to  w  of  the  firrt  integral. 


344  Equations  in  Three  Variables 

where  A  =  W/ZK.  A  fundamental  solution  of  this  equation  is  given  by 

0  = 


where       R2  =  (x  -  £)2  -f  (y  -  7?)2  +  (z  -  £)2,     (&  *?,  £  constant). 

In  particular,  if  £  =  77  =  £  =  0,  we  have  the  solution  0  =  Ar~le~^r, 
where  r  is  the  distance  from  the  origin,  and  this  corresponds  to  the  solution 

6=  Ar-le*(*-r\  ......  (T) 

This  solution  has  been  used  by  H.  A.  Wilson*  and  H.  Machef  to  account 
for  the  following  phenomenon. 

If  a  bead  of  easily  fusible  glass  (0)  be  placed  a  few  millimetres  above 
the  tip  of  the  inner  cone  (K)  of  the  flame  of  a  Bunsen  burner,  a  sharply 
defined  yellow  space  (SSf)  of  luminous  sodium  vapour  is  formed  in  the 

current  of  gas  which  is  ascending  vertically 
with  considerable  velocity.  This  space  envelops 
the  bead  and  broadens  out  in  the  higher  part 
of  the  flame,  as  shown  in  Fig.  27.  Provided 
the  gas-pressure  is  not  too  high,  the  critical 
velocity  of  Osborne  Reynolds,  at  which 
turbulence  sets  in,  will  not  be  exceeded  even 
in  these  parts  of  the  flame,  so  that  the  flow 
remains  laminar,  and  the  sodium  vapour  de- 
veloped from  the  bead  is  driven  into  the  hot 
gas  solely  under  the  influence  of  diffusion. 

The  fact  that  the  vapour  extends  beneath 
the  bead  in  the  direction  OA  is  proof  of  the 
high  values  of  the  coefficient  of  diffusion 
assumed  at  high  temperatures,  and  at  this 
point  diffusion  must  be  able  to  more  than 
counteract  the  upward  flow.  Since  an  iso- 
thermal surface  corresponds  in  the  theory  of  diffusion  to  a  surface  of  equal 
partial  pressure,  it  is  supposed  that  for  suitable  constant  values  of  A  and 
8  the  equation  (T)  represents  the  surface  enclosing  the  sodium  vapour 
developed  from  the  glass  bead.  When  K  is  small  and  w  large,  this  surface 
approximates  to  the  form  of  a  paraboloid  of  revolution  with  the  origin  as 
focus. 

Mache  obtains  the  solution  by  integrating  the  effect  of  an  instantaneous 
source  which  is  successively  at  the  different  positions  of  a  point  moving 
relative  to  the  medium  with  velocity  w.  In  fact 


Fig.  27. 


j-O 

)-i 
Jo 


where  A  =  w/2/c. 

*  Phil.  Mag.  (6),  vol.  xxiv,  p.  118  (1912);  Proc.  Camb.  Phil  Soc.  vol.  xu,  p.  406  (1904). 
|  Phil.  Mag.  (6),  vol.  XLVII,  p.  724  (1924). 


Diffusion  of  Smoke  345 

A  similar  solution  has  been  used  by  O.  F.  T.  Roberts*  to  give  the 
distribution  of  density  in  a  smoke  cloud  when  the  smoke  is  produced 
continuously  at  one  point,  and  at  a  constant  rate.  The  case  in  which  the 
smoke  is  produced  continuously  along  a  horizontal  line  at  right  angles  to 
the  direction  of  the  wind  is  solved  by  integrating  the  solution  for  the 
previous  case. 

§  5-  31.  Two-dimensional  motion  of  a  viscous  fluid.  If  (u,  v)  are  the 
component  velocities  at  the  point  (x,  y)  at  time  t,  p  the  pressure  at  this 
point,  the  equations  of  motion,  when  the  fluid  is  incompressible  and  of 
uniform  density  />,  are 

du          du          du          1  dp 

+  u        +  v       =  --  _£ 

ot  ox  oy  p  ox 

dv  dv  dv  1  dp        _ 

+  u+v==  --  _*:  +  v  V2?;  , 

ot          ox          oy  p  oy 

while  the  equation  of  continuity  is 


This  last  equation  may  be  satisfied  by  writing 

ddf  dJj 

<jj    -  '  ,j/    _  _  T_ 

U    -     <~\        )  ^    -  ~f\~   j 

dx  dy 

where  if*  is  the  stream-function,  and  if 

fccfc 

dx      dy          r 
is  the  vorticity  at  the  point  (x,  y)  at  time  t,  we  have 


or 


If  s  —  xv  —  yu,  we  have 

ds     dv    du          Ss    dv    du 
~-  =  x  x  --  v  ^-  +  VJ      ^  =  a;  -^  --  y  x~  — 
9x    dx   y  dx          dy    dy   y  dy 

d*s  __      y*y  _      d*u          dv       d*s  _      92y          9 

9^2  -  *  9^1  -  y  a^2  +  2  g^'    3^2  -  x 

TT  ^5  35  ds  1  /     3p  9^ 

Hence  -^  +  ^5-+  v  5-  =  --  (x  -*r  ~  2/^" 
3^          3a;          3y          p  \    dy      y  ox 

If  x2  +  y2  =  r2  we  may  write 

ds         dv  ds  du 


ds         ds         f    dv          du\ 

u  o~+  v  5"  =  M  w  a  --   V-^~. 

3x        3t/        V    3r          9r/ 
*  Proc.  jRoy.  Soc.  London,  vol.  orv,  p.  640  (1923). 


346  Equations  in  Three  Variables 

If  the  flow  is  of  such  a  nature  that  p  depends  only  on  r  and  v/u  is 
independent  of  r,  we  have 


o-  u 

since  s  =  r  vx  ,  we  have 

dr 


i 

.  _r  —  r        /  __ 
~  r 


Hence  in  the  special  case  when  ^  depends  only  on  r,  and  the  velocity 
is  everywhere  perpendicular  to  the  radius  from  the  origin,  we  have  the 
d.  ^rential  equation 


This  indicates  that  the  velocity  V  =  s/r  satisfies  the  equation 


j*-Tj-fo- 

which  is  of  the  same  form  as  the  equation  of  the  conduction  of  heat  when 
the  temperature  0  is  of  the  form  0  =  V  cos  6. 

In  the  present  case  if/  and  £  are  related  since  they  both  depend  on  r 
and  so  the  equation  for  £  is 


The  equation  satisfied  by  ^  is 


where  /  (t)  is  an  arbitrary  function  of  t. 
In  the  particular  case  when 


we  have  s  -  -  (r2/2v^2)  e-f2/4*«,     F  =  -  (r/ 

The  total  angular  momentum  is  in  this  case 

TOO 

srdr  =  — 


o 
and  is  constant.  The  kinetic  energy  is  on  the  other  hand 

TTO  (^  V*rdr  =  Trp/2^2. 
Jo 

This  type  of  vortex  motion  has  been  discussed  by  G.  I.  Taylor*  in 
connection  with  the  decay  of  eddies.  The  corresponding  type  of  vortex 
motion  in  which  r  _.  j-i  e-r«/4v< 

ias  been  discussed  by  Oseenf,  TerazawaJ  and  Levy§. 

*  Technical  Report,  Advisory  Committee  for  Aeronautics,  vol.  I,  1918-19,  p.  73. 

t  C.  W.  Oseen,  Arkiv  f.  Mat.,  Astr.  o.  Fys.  Bd.  vn  (1911). 

J  K.  Terazawa,  Report  Aer,  Res.  Inst.,  Tokyo  Imp.  Univ.  (1922). 

§  H.  Levy,  PhiL  Mag.  (7),  vol.  n,  p.  844  (1926). 


Decaying  and  Growing  Disturbances  347 

§  5-32.    Solutions  of  the  form  iff  =  X  (x,  t)  +  Y  (y,  t).  The  condition  to 
be  satisfied  is 


_ 
dy*dt      dx  lty»      dy  dx*  ~ 

Differentiating  successively  with  respect  to  x  and  y  we  get 

d2X^Y_d*Yd*X^ 
dx*  dy*       dy*  dx* 

We  can  satisfy  this  equation  either  by  writing 


X  =  xa'  (t)  +  b  (t),     Y  =  yA'  (t)  +  B  (t),          ......  (B) 

,         ...  d'X      r    U^&X      d*Y      r     mi,92y  /p. 

or  by  wnting       -^  =  [/t  (f)]»  ^  ,     ^  =  [M  (*)]•  ^  .......  (C) 

The  supposition 

(D) 
......  ^ 


11^  A  A 

leads  to  -^---  =  0,      -~T  =  0. 

3x4  3^/4 

These  equations  follow  from  (C)  if  we  put  /u,  (t)  =  0. 
Solving  equations  (C)  for  X  and  7  we  get 

X  =  a(t)  e**M  +  b  (t)  er*»™  +  xc  (t)  +  d(t), 
Y  -  A  (t)  e^«>  4-  B  (t)  e-y*w  +  yC  (t)  +  D  (t). 

Substituting  in  the  original  equation  and  assuming  that  a  (t),  b  (t), 
A  (t),  B  (t)  are  not  zero,  we  find  that  /x  (t)  must  be  a  constant  ^  and  that 
the  functions  a,  6,  c,  A,  B,  G  must  satisfy  the  equations 

f  -  Cap,*  =  vap,*,     ^b'  -f 


primes  denoting  differentiations  with  respect  to  t. 

If  the  functions  c  (t)  and  C  (t)  are  chosen  arbitrarily,  a(f),  b  (t),  A  (t) 
and  B  (t)  may  be  determined  by  means  of  these  equations  when  their 
initial  values  are  given.  In  particular,  if  a  =  A  =  0,  c  =  C  =  0,  we  can  have 

6  -  Pe1**,    B  = 


u  =  p,yeVf*'*r~ILV9     v  =  — 
This  represents  a  growing  disturbance  in  which  each  velocity  com- 
ponent is  propagated  like  a  plane  wave.  The  pressure  is  given  by  the 
equation  r/J/n 

F+J4_0  +  1 


348  Equations  in  Three  Variables 

The  fluid  may  be  supposed  to  occupy  the  region  x  >  0,  y  >  0.  If  so, 
fluid  enters  this  region  across  the  plane  x  =  0  (Q  >  0)  and  leaves  it  at  the 
plane  y  =  0  (P  >  0).  The  amount  entering  the  region  is  equal  to  the  amount 
leaving  the  region  if  P  =  Q,  the  density  p  being  assumed  constant. 

If  V  =  0  and  pn  is  the  pressure  at  infinity  (a:  =  oo,  y  =  oo),  we  have 
P  -  ^oo  =  pfji*P*e2»2vt-*(x+v). 

The  pressure  is  generally  greater  than  pm  and  is  propagated  like  a 
plane  wave  with  velocity 

c  ==  iiv  A/2  =  v  —  -  —  . 
^  u  —  v 


Thus  the  velocity  of  a  plane  pressure  wave  in  an  incompressible  fluid 
is  equal  to  v  times  the  ratio  of  the  vorticity  and  the  transverse  component 
of  velocity. 

When  the  motion  is  steady  the  equation  to  be  satisfied  is 
aer*  (VIL  -f  C)  +  be-**  (vji  -  C)  +  Aer*  (vp>  -  c)  +  Be~™  (vp,  +  c)  =  0, 
and  we  have  four  typical  solutions  : 

0  =  jps2  +  ex  +  qy*  +  Cy  +  D, 
ifj  =  v^y  +  be-**  +  ex  +  d, 
iff  =  VIJL  (x  +  y)  -f  Ae,™  -f-  be~*x  +  d, 
i/j  =  Aer*  -f  vpx  -f  Cy  +  D. 

^2  \r 

Returning  to  the  first  case  we  note  that  when  -~  2  =  0  the  equations 
(B)  do  not  give  all  possible  solutions,  for  if 

X  -  xa'  (t)  -f-  6  (0, 
the  original  equation  becomes 


Writing  C7  =  -=—  2-  we  have  the  simpler  equation 


which  possesses  a  solution  of  type 

£  s  U  -  f  V"*2'  cos  A  [y  -  a  (t)]  o>  (A)  d\, 
Jo 

where  o>  (A)  is  a  suitable  arbitrary  function.  For  the  corresponding  motion 

roo  fJ\ 

4>  =  aw'  («)  -f  yc  (*)  +  6  (<)  +       c-*"  cos  \\y-a  (t)}  <a  (A)  -~, 

Jo  A* 

r°°  ^/A 

«  -  c  (t)  -        c-'«  sin  A  [y  -  a  (<)]  w  (A)  ^, 
Jo  A 


v  - 


Laminar  Motion  349 

This  solution  may  be  used  to  study  laminar  motion.  The  corresponding 
solution  for  the  case  in  which  the  motion  is  steady  is 

*? 

i/i  -  Kx  +  Pe  v  +  #7/2  +  Ry  +  flf, 

where  P,Q,  It,  S,  K  are  arbitrary  constants.  If  J£  ->  0  while  the  coefficients 
P,  Q,  /?,  $  become  infinite  in  a  suitable  manner,  a  limiting  form  of  the 
solution  gives  the  well-known  solution 

0  -  Ay*  +  Qy*  +  Ry  +  S. 

It  may  be  mentioned  here  that  an  attempt  to  find  a  stream-function  iff 
depending  on  a  parameter  s  but  not  on  t,  and  such  that 


1     ,14-        *U  *•  //  X 

led  to  the  equation          ^  -  ^  =  /  (a;,  j,) 

The  conditions  for  the  compatibility  of  this  equation  and 

' 


seem  to  require  /  (#,  y)  to  be  a  constant.   By  a  suitable  choice  of  axes  the 
former  equation  may  then  be  reduced  to  the  form 


_  n 
dxdy~    ' 

and  so  0  =  JT  (x,  s)  4-  7  ( 


EXAMPLES 

1.  In  the  case  when  there  is  a  radial  velocity  U  and  a  transverse  velocity  F,  both  of 
which  depend  only  on  r  and  t  and  when  the  pressure  p  depends  only  on  r  and  t,  the  equations 
for  U  and  F  are 


dV      TrdV      UV         (d2F       1  dV       1  T71   3 

a*'  +  ^  a~  ^  ---  =  v  1^~2   +  ~    3     ~  ~2  FT  a" 
5^  dr         r  (  dr*       r    dr       r2     )  dr 

Hence  show  that  F  satisfies  the  equation 


F 
~ 


where  ^L  is  a  constant.  If  a  —  X/2v,  prove  that  there  is  a  solution  of  type 

y  ^r2<T+it-«-2e-r2/*vt9 
and  verify  that  the  total  angular  momentum  about  the  origin  remains  constant. 

2.   Prove  that  the  equation  for  F  is  satisfied  by  a  series  of  type 

y  _  rn*-m  1  1  __      _  .    ,  _  ™.  ____  (r2/vt} 
V      rt       \        (1  +  »  -  2a)  (n  +  3)  (T  /vl> 

_ 
v   '    ;  ' 


350  Equations  in  Three  Variables 

and  verify  that  when  n  =  1,  m  =  2, 


rf-'e-  <«•"«»»  (l  +  --"-   (r*/M)  +  -^    1  (rV4rf)*  +  ...1. 

^  CT  —    I  (7    —   &     41  J 


This  is  a  particular  case  of  Kummer's  identity  F  (a;  y;  a;)e"a:=  ^(y  —  a;  y;  ~^),  where 
F  (a;  y;  ar)  is  the  confluent  hypergeometric  function  (Ch.  ix). 

3.   Prove  that  there  is  a  type  of  two-dimensional  flow  in  which 

{  =  *V, 
and  0  is  consequently  of  the  form 

t  =  €-****  F  (x,  y), 
where  F  (x,  y)  satisfies  the  differential  equation 


A 
=  0. 


Prove  that  in  the  latter  case  if  a2  >  62  there  is  a  growing  disturbance  which  is  propagated 
with  velocity  vk2/a,  and  show  that 


Discuss  the  cases  in  which 

F  =  cos  ax  cos 


a  u' 


CHAPTER  VI 

POLAR  CO-ORDINATES 

§  6-11.  The  elementary  solutions.  If  we  make  the  transformation 

x  =  r  sin  0  cos  <£,     y  =  r  sin  0  sin  <£,     2  =  r  cos  0, 
the  wave-equation  becomes 
&W     2dW          1        3 


/  .        3]f\  _!  __          _ 

"3r«       r  ~3r  "     r2  sin  0  30  \sm      30  )  +  r2  sin2  0  9e/>2      c2    9*2   ~ 
This  is  satisfied  by  a  product  of  type 

W=R(r)®(0)Q(<f>)T(t),  ......  (I) 

Mm 
if  +  WcT  =  0, 


sin  i 


ar*      r  ar 

where  k,  m  and  n  are  constants. 
The  first  equation  is  satisfied  by 

T  =  a  cos  (to)  4-  6  sin 
where  a  and  6  are  arbitrary  constants ;  the  second  equation  is  satisfied  by 

O  =  A  cos  m<f)  4-  J5  sin  m<£, 

where  .4  and  B  are  arbitrary  constants.  The  third  equation  is  reduced  by 
the  substitution  cos  0  =  p,  to  the  form 


Its    solution    can   be   expressed  in    terms    of    the  associated  Legendre 
functions  Pnm  (/x)  and  Qnm  (p)  which  will  be  defined  presently. 

When  k  =  0  the  fourth  equation  has  the  two  independent  solutions  rn 
and  r~(n+1),  except  in  the  special  case  when  n  =  —  (n  4-  1),  i.e.  when 
n  =  —  £.  Making  the  substitution  w  =  r%  R  in  this  case  we  obtain  the 

equation 

d*w     1  dw  _ 
dr^^r  dr  ~  U' 

which  is  satisfied  by  w  ==  C  +  I>  log  r,  where  (7  and  Z>  are  arbitrary 
constants. 


352  Polar  Co-ordinates 

The  fact  that  rn  and  r"n~l  are  solutions  of  the  equation  for  R  furnishes 
us  with  an  illustration  of  Kelvin's  theorem  that  if  /  (x,  y,  z)  is  a  solution 
of  Laplace's  equation,  then 

1  f(x  y,  ?\ 

r  J\r*>  r2'  r*) 

is  also  a  solution.  The  transformation  in  fact  transforms  rn  0<1>  into  r~n~l  0O ; 
it  also  transforms  r~i  (G  -f  D  log  r)  Q<J>  into  r~*  (G  —  D  log  r)  0<I>. 

When  m  =  0  and  n  =  0  the  differential  equation  for  0  is  satisfied  by 


Thus,  in  addition  to  the  potential  functions  1  and  -,  we  have  the 

potential  functions 

1  !      r  +  z        ill      r  +  z 
2lo8r_i   and   2rlogF-V 

It  should  be  noticed  that 

3  /I.      r  +  z\       1 


1  1 

In  fact  we  have  ^-  log  (r  +  z)  =  -  , 


and  it  is  easily  verified  that  log  (r  -f  z)  and  log  (r  —  z)  are  solutions  of 
Laplace's  equation.  These  formulae  are  all  illustrations  of  the  theorem  that 
if  W  is  a  solution  of  Laplace's  equation  (or  of  the  wave-equation),  then 


is  also  a  solution  of  Laplace's  equation  (or  of  the  wave-equation). 

§  6-12.   In  the  case  of  the  wave-equation  the  solution  corresponding  to 
l/r  is  etkr/r,  and  there  are  associated  wave-functions 

-  cos  k  (r  —  ct),     -  sin  k  (r  —  ct), 
which  are,  of  course,  particular  cases  of  the  wave-function 


in  which /(T)  is  an  arbitrary  function  which  is  continuous  (D,  2). 

§  6-13.  In  the  case  of  the  conduction  of  heat  the  fundamental  equation 
possesses  solutions  of  the  form  (I)  where  R,  0,  O  satisfy  the  same 
differential  equations  as  before  but  T  is  of  type 

a  exp  (- 


Cooling  of  a  Spherical  Solid  353 

where  a  is  a  constant  and  h2  is  the  diffusivity.  Thus  there  are  solutions 
of  type 


-  cos  kr  .  e~w  ,     -  sin  kr  .  e- 


which  depend  only  on  r  and  t.  The  second  of  these  is  the  one  suitable  for 
the  solution  of  problems  relating  to  a  solid  sphere.  If,  in  particular,  there 
is  heat  generated  at  a  uniform  rate  in  the  interior  of  the  sphere  the 
differential  equation  for  the  temperature  6  is 


ot 
where  6  is  a  constant.  There  is  now  a  particular  integral  —  b*r2/6h2  which 

r)f) 

must  be  added  to  a  solution  of  -^  =  h2V29. 

ct 

If  initially  6  *=  00  throughout  the  sphere,  60  being  a  constant,  and  the 
boundary  r  =  a  is  suddenly  maintained  at  temperature  01  from  the  time 
t  =  0  to  a  sufficiently  great  time  T,  the  condition  at  the  surface  is  satisfied 
by  writing 


while  the  initial  condition  is  satisfied  by  writing* 


_ 

As  £  ->  oo,  0  tends  to  the  value  0t  +  —  -  •  .  2  --  ;  and  ^-  to  the  value 

62r 
—  ^r-,  so  that  the  flow  of  heat  across  the  surface  is,  per  second, 


O  K 

Writing  62  =  —  and  h2  =  —  ,  where  p  is  the  density  and  a  the  specific 

heat  of  the  substance,  we  have  the  result  that  the  rate  of  flow  of  heat 
across  the  surface  is  4Q7ra3/3,  a  result  to  be  anticipated. 


_ 

If,  on  the  other  hand,  the  initial  temperature  is  0t  -\  --  7^2 

the  surface  of  the  sphere  radiates  heat  to  a  surrounding  medium  at 
temperature  02  at  a  rate  E  (9a  —  02)  per  square  centimetre,  where  6a  is  the 
(variable)  surface  temperature  of  the  surface  of  the  sphere,  the  solution  is 


e  -  B  -  ~  +  -  XDne~n2M  sin  nr. 
on*      r 

*  The  constant  Dm  is  obtained  by  Fourier's  rule  from  the  expansion  of  00  -  6l  -  ^  2  (a2  -  ra) 
in  a  sine  series. 

B  2 


354  Po/ar  Co-ordinates 

The  surface  condition  is  satisfied  by  writing 

R  -  fl    4-  a6*  (^  + 

2  + 


where  <pm  is  the  with  root  of  the  transcendental  equation 


The  initial  condition  gives 


Fr  =  £  Dn  sin  nr, 

m-l 


where 

and  the  extended  form  of  Fourier's  rule  gives 

z-  K)' 
Ea  -  E 

(Ea  - 


2F   K2<l>m2  4-  (Ea  -  K}*    f«       .     (r^m\  3 
I~T~W~TF W\      r-sm  I        )  dr 


These  results  have  been  used  by  J.  H.  Awbery*  in  a  discussion  of  the 
cooling  of  apples  when  in  cold  storage. 

§  6-21.   Legendre  functions.   The  method  of  differentiation  will  now  be 
used  to  derive  new  solutions  of  Laplace's  equation  from  the  fundamental 

i   ,  .         1       ,  1  ,      r  +  z 
solutions  -  and  ^-  log  -  . 
r          2r     &  r  -  z 

After  differentiating  n  times  with  respect  to  z  the  new  functions  are  of 
form  r-n~lQ;  consequently  we  write 


n 


i      Lt  2 
g     - 


''n^>-  n\  dz«\2r^r-z) 
and  we  shall  adopt  these  equations  as  definitions  of  the  functions  Pn  (p.) 
and  Qn  (p)  for  the  case  when  n  is  a  positive  integer  and  6  is  a  real  angle. 
The  first  equation  indicates  that  there  is  an  expansion  of  type 

(^•2  tynfii     I     /Tf2\    j  — —     \*       —    -    ~P     ( ii\          I  n   I    *?*    I  Y 
^ti-/  fJi     i~   i4/i         —     ^j      />«w+l  •*•   n  \r^)i  I    ^^  I 

n— 0 

and  this  equation  may  be  used  to  obtain  various  expansions  for  Pn(fi).  Thus 
1.3  ...  (2n-  1) 


3»^=         1.2..., 


V-+...] 


- ~       =  (-)-  F    -  n,  n 


.  (7),  vol.  iv,  p.  629  (1927). 


Hobsorfs  Theorem  355 

where  F  (a,  6;  c;  x)  denotes  the  hypergeometric  series 

1  +  a'bx  .  M«Ji!L6J*±JQ  *;2  , 
+  l.c*4"      f.2.c(c  +  1)  "" 

§  6-22.  Hobsorfs  theorem.  The  first  expansion  for  Pn  (^)  is  a  particular 
case  of  a  general  expansion  given  by  E.  W.  Hobson*.  If  /  (x,  y,  z)  is  a 
homogeneous  polynomial  of  the  nth  degree  in  x,  y,  z, 


r2V2  r*V4 

X 


2  (2?)       2.4 
When  /  (x,  y,  z)  =  zn  this  becomes 


1  , 
^3)       '"J  ;  (*J  y'  Z)' 


r  n  (n  -  1)  2      n  (»  -J)Jn  -_2)  Jn  -3)          4  _      1 

A  1Z        2  (2/1  -  :  1)     z       +     274  (271  -  1)  (2n  -  3)  "'J' 

which  is  equivalent  to  the  expansion  for  n  I  r~n~lPn  (^). 

Assuming  that  the  theorem  is  true  for  /  (x,  y,  z)  =  zn  it  is  easy  to  see 
that  the  theorem  must  -also  be  true  for  /  (x,  y,  z}  =  (£x  -f  r^y  +  ^)n,  where 
|x  +  77^  +  ^  is  derived  from  z  by  a  transformation  of  rectangular  axes,  for 

r\  /"\  ^\  ^\ 

such  a  transformation  transforms  ^-  into  ^  ^    -f  -n  ^-  +  t  «-    and  leaves 

S^  ra        '  cy          dz 

V2  unaltered. 

To  prove  that  the  theorem  is  true  in  general  it  is  only  necessary  to 
show  that  /  (x,  y,  z)  can  be  expressed  in  the  form 

/  (x9  y,z)=   2  As  ($8x  +  w  +  £,*)», 

s=l 

where  the  coefficients  As  are  constants. 

To  determine  such  a  relation  we  choose  k  points  such  that  they  do  not 
all  lie  on  a  curve  of  degree  n  and  such  that  a  curve  of  degree  n  can  be  drawn 
through  the  remaining  k  —  I  points  when  any  one  of  the  group  of  k  points 
is  omitted.  Let  £s,  rjs,  £s  be  proportional  to  the  homogeneous  co-ordinates 
of  the  sth  point  and  let  i/js  (x,  y,  z)  =  0  be  the  equation  of  the  curve  of 
degree  n  which  passes  through  the  remaining  k  —  1  points. 

Assuming  that  a  relation  of  the  desired  type  exists  we  operate  on  both 

sides  of  the  equation  with  the  operator  i/j8  (  ^—  ,  ^—  ,  ^  -  j  .    The  result  is 

'  ~dy' 

Giving  s  the  values  1,  2,  ...  k  all  the  coefficients  are  determined.  Since 
a  curve  of  the  nth  degree  can  be  drawn  through  \n  (n  -f-  3)  arbitrary  points, 

*  Proc.  Land.  Math.  Soc.  (1),  vol.  xxiv,  p.  55  (1892-3). 

23-2 


356  Polar  Co-ordinates 

the  number  k  should  be  taken  to  be  \  (n  +  1)  (n  +  2),  which  is  exactly  the 
number  of  terms  in  the.  general  homogeneous  polynomial  /  (#,  y,  z)  of 
degree  n.  The  coefficients  A3  could,  of  course,  be  obtained  by  equating 
coefficients  of  the  different  products  xaybzc  and  solving  the  resulting  linear 
equations,  but  it  is  not  evident  a  priori  that  the  determinant  of  this  system 
of  linear  equations  is  different  from  zero.  The  foregoing  argument  shows 
that  with  our  special  choice  of  the  quantities  gs  ,  77  5  ,  £8  the  determinant  is 
indeed  different  from  zero  because  with  a  special  choice  of  /,  say 


the  equations  can  be  solved. 

The  solution  is,  moreover,  unique  because  if  there  were  an  identical 
relation 

0  =  S  Cs  (tsx  +  w  +  £,*)», 

3=1 

the  foregoing  argument  would  give 

O^nlC^tf,,^,  £.). 

Hobson's  theorem  has  been  generalised  so  as  to  be  applicable  to 
Laplace's  equation  for  a  Euclidean  space  of  m  dimensions.  Writing 

V  .  =  -£+^+       +     ** 
m   -^^   •"   [  ' 


and  using/  (^  ,  a;2,  ...  xm)  to  denote  a  homogeneous  polynomial  of  degree  n, 
the.  general  relation  is 

\'  4'  "'  9 


[r'2Vm2 
1  ~  2  (m  +  2n  -  4)  +  ^ 


2n  -  6) 

§  6-23.   Potential  functions  of  degree  zero.   When  n  =  0  the  differential 
equation  satisfied  by  the  product  U  =  0O  may  be  written  in  the  form 

a2  17 


_ 

~  ' 

U  f          ^  f        <W  1         -4.  ^ 

where  s  =  -     -,—  ^—  9  =     ~^  —  5  ^  log.tan  -  . 

)  I  ~  p.2     ]   smO         &i       2 

It  follows  that  there  are  solutions  of  type 


where  /is  an  arbitrary  function  and/  (u)  = 
This  solution  may  be  written  in  the  form 


Differentiation  of  Primitive  Solutions  357 

where  F  is  an  arbitrary  function.  The  general  solution  of  Laplace's  equation 
of  degree  zero  may  thus  be  written  in  the  form 


r 


where  F  and  G  are  arbitrary  functions*.  The  general  solution  of  degree  —  1 
may  be  obtained  from  this  by  inversion  and  is 

\  „  (x  —  ii 


-  0 


>r  +  zj     r      \r  +  z. 

Solutions  of  degree  —  (n  4-  1)  may  be  obtained  from  the  last  solution 
by  differentiation.   In  particular,  there  is  a  potential  function  of  type 


^   H  ix  +  i 
~~  8z»  [r\z  +  r 


which  is  of  the  form  r-"-1^  (0)  em*.  The  function  must  consequently  be 
expressible  in  terms  of  Legendre  functions.  When  m  is  a  positive  integer 
equal  to  or  less  than  n  we  have  in  fact  the  formula  of  Hobson 


)n  n  /T  .L  •»?Amn 

(/  (ITT)  J  =  {~}n  (n  ~  m)  !  r"n"lp» 


When  m  is  a  positive  integer  greater  than  or  equal  to  n  we  have  the 
expansion 

fl    ^        (9-n        1\        1     **        (9n        ^\  9m        1 
,       .      \   L  .  O  . . .  \LTl  —    1)          L  .  o  . . .  I  4  /I  —   *>  l  £i  71  —    1 ,  . 

:  V       )     I  ~2V~1  /      rn^n  ~^  ~2n~? m~n    I  1 (m  ~~  n) 

+  r2n-I7^  rTm-nV2   ~~ j    2 (w  -  7l)  (m  -  ?*  +   1)  +   ... 

...  4-  — _j —  (m  —  n)  (m  —  n+  1)  ...  (m  — 

which  may  be  used  to  define  the  function  x  (0) in  this  case.  In  particular, 
we  have  the  relation 

an_  r  ___! 

dzn  [r  (z  -f  r 

When  this  is  used  to  transform  the  expression  for  r~n-lPn" 
we  find  that| 


*  W.  F.  Donkin,  Phil.  Trans.  (1857). 

t  This  formula  is  given  substantially  by  B.  W.  Hobson,  Proc.  London  Math.  Soc.  (1),  vol.  xxn, 
p.  442  (1891).  Some  other  expressions  for  the  Legendre  functions  are  given  by  Hobson  in  the 
article  on  "Spherical  Harmonics"  in  the  Encyclopedia  Britannica,  llth  edition. 


358  Polar  Co-ordinates 

EXAMPLE 

Prove  that  if  m  is  a  positive  integer 


m  pi 
?r  J  0 

T2 
/ 

T  J  o 


,  elmada 

z  +  ix  cos  a  -f  ii/  sin  a    r  \z  +  r 

\  (  x 

— 

z 


i      ,         .  .     .       v    ,• 

log  (z  -f  ix  cos  a  +  it/  sin  a)  etmada  =  -  -    —  r  ~~ 

-\-  r  / 


§  6-24.    Upper  and  lower  bounds  for  the  function  Pn  (/x).   We  shall  now 
show  that  when  —  1  <  /x  <  1  the  function  Pn  (/x)  lies  between  —  1  and  -f  1. 
This  may  be  proved  with  the  aid  of  the  expansion 

— --" cos  n>8  -f-  A  .    -~  -  *        -  — -  cos  (n  —  2)  6 

1.3   1.3  ...  (2n-  5) 

+    n 


2.4-2.4...  (2n-4rwv"         '      '   ' 
which  is  obtained  by  writing 

(1  -  2x  cos  9  -f  x2)~i  -  (1  -  xe*)-*  (1  -  ze-'T** 

and  expanding  each  factor  in  ascending  powers  of  x  by  the  binomial 
theorem,  assuming  that  |  x  \  <  1. 

It  should  be  observed  that  each  coefficient  in  the  expansion  is  positive, 
consequently  Pn  (/*)  has  its  greatest  value  when  6=0  and  /x,  =  1,  for  then 
each  cosine  is  unity. 

If,  on  the  other  hand,  we  replace  each  cosine  by  —  1,  we  obtain  a 
quantity  which  is  certainly  not  greater  than  Pn  (/x).  Hence  we  have  the 
inequality  -  i  <  P.  (/t>  <  i,  fc*  _  1  <  ,.  <  1. 

When  n  is  an  odd  integer  Pn  (/x)  takes  all  values  between  —  1  and  +  1, 
but  when  n  is  an  even  integer  Pn  (ju)  has  a  minimum  value  which  is  not 
equal  to  —  1.  This  minimum  value  is  —  |  for  P2  (p)  and  —  f  for  P4  (/x). 

§  6-25.  Expressions  for  the  Legendre  polynomials  as  nth  derivatives. 
Lagrange's  expansion  theorem  tells  us  that  if 

z  =--  IJL  +  acf)  (z), 

the  Taylor  expansion  of/'  (z)  -v-  in  powers  of  a  is  of  type 

(tjJL 


az  =     -       - 


1  . 

a2)*,       -  =  (1  -  2/xa  +  a2)-*; 


Formulae  of  Rodrigues  and  Conivay  359 

a  comparison  of  coefficients  in  this  expansion  and  the  expansion 

(1  -  2/ta-f  a2)-*  =  S  a»Pn  (M) 
o 

gives  us  the  formula  of  Rodrigues, 

p«  ^  -  2-4l  £-  [("3  ~  1)nj- 

If,  on  the  other  hand,  we  write 

c£  (z)  -  2  (V~z  -  0, 
we  have  3  =  /z  4-  2a  (Vz  —  t)9 

z  -  2a  Vz  +  a2  =  a2  -  2a^  +  ti, 


=  a  ±     a2  —  2a^  -f  M, 

n  +  1 


/  M  =  ?"  _31 

\vV/     n !  3/xn       vV 
1       cn     (r  —  t)n 


Hence 


or  ....- ,-t  *  »  i :.  i      ~\(rdr)n       r 

This  formula  is  due  to  A.  W.  Conway,  the  previous  one  to  E.  Laguerre. 
Replacing  t  by  z  we  have  the  following  expression  for  a  zonal  harmonic 

1     „    ,   .        1      3n      (^-^Y 

r       ' 


z  and  r  being  regarded  as  independent. 

§6-26.  The  associated  Legendre  functions.  The  differential  equation  (II) 

m 

of  §  6-11  is  transformed  by  the  substitution  0  =  (1  —  p,2)2  P  to  the  form 


but  this  equation  is  satisfied  by  P  =  j-~  ,  where  v  is  a  solution  of  Legendre's 

equation 

,  //27)  fin 

(I-/**)  ^,-2/*;fc+»<»+l)'-0, 
particular  solutions  of  which  are  Pn  (/x)  and  Qn  (/x). 


360  Polar  Co-ordinates 

Hence  we  adopt  as  our  definitions  of  the  functions  Pnm  (/z)  and  Qn™  (/x) 
for  positive  integral  values  of  n  and  m 


m 

o    nm 


-  1  <  /x<  1. 

With  the  aid  of  these  equations  we  may  obtain  the  difference  equations 
satisfied  by  Pnm  (p.)  and  Qnm  (//,)  : 

(TI  -  m  4-  1)  Pwr?+1  -  (2n  4-  1)  /zPn™  +  (n  +  m)  Pmn^  =  0, 


+l  =  2mp.Pnm  -  (n  +  r/i)  (n  -  m  +  1)  Vl 
P"*,,.,  -  jzPn™  -  (n  -  m  +  1)  Vl  - 
Pmn+1  -  /xPww  4-  (n  4-  w 


1  -  (n  4-  m  4-  1)  fi.Pnm  -  (n  -  m  4-  1)  P^+i, 
and  the  following  expressions  for  the  derivative 

(I-/*')  ^  Pn»  (/,)  -  (n  +  1)  /JV»  (/z)  -  (^  -  m  +  1)  P-n,!  (/x) 
-  (w  +  w)  P-w_i  (/x)  -  n/iPn~  (/x). 

Similar  expressions  hold  for  the  derivative  of  Qnw  (/*)• 

Expressions  for  the  Legendre  functions  of  different  order  and  degree  n 
are  easily  obtained  from  the  difference  equations  or  from  the  original 
definitions.  In  particular 

P0°=  1- 

P!°  -  cos  0,     P^  -  sin  6. 

P2°  =  i  (3  cos2  19-1),     Pa1  -  3  sin  9  cos  0,     P22  -  3  sin2  5. 

P3°  -  \  (5  cos3  0-3  cos  0),     P31  -  |  sin  0  (15  cos2  0  -  3), 

P32  -  15  sin2  9  cos  9,     P33  -  15  sin3  0. 

P4°  -  I  (35  cos4  0-30  cos2  9  4-  3),     P^  -  J  sin  0  (35  cos3  0-15  cos  0), 

P42  -  £  sin2  0  (105  cos3  0  -  15),     P43  -  105  sin3  0  cos  0,     P44  -  105  sin4  0. 

P6o  =  j  (63  cos5  0-70  cos3  04-15  cos  0), 

Pg1  -  I  sin  0  (315  cos4  0  -  210  cos2  0  4-  15), 

P62  -  \  sin2  0  (315  cos3  0  -  105  cos  0), 

P63  ^  |  Sin3  ^  (945  Cos2  0  _  105), 

P64  -  945  sin4  0  cos  0,     P55  -  945  sin5  0. 


A  ssociated  Legendre  Functions  361 


EXAMPLES 
1.  Prove  that  if  m  and  n  are  positive  integers 


2.  Prove  that 


ai  ~~  *  cf")  ^  Pn?W  (/i)  6*m^       =•  (»  +  m)  (w  +  w  -  1)  rn~l  Pn 
(jx  +  id]  [r""n""1  Pr*m  (fz)  ^^  =  ""  r~n~2  Pn+lWl 

(w  ~  m  +  1J  (n  ~ 


§  6-27.  Extensions  of  the  formulae  of  Rodrigues  and  Conway.  By 
differentiating  the  formula  of  Rodrigues  m  times  with  respect  to  ^  we 
obtain  the  formula 


We  shall  use  a  similar  definition  for  negative  integral  values  of  m  and 
shall  write 


Expanding  by  Leibnitz's  theorem  we  obtain 


n—  m  (n  __  w\  f  72,  1 

_     V  VAf>         AA6^  '  __  _ 

~ 


Comparing  the  two  series,  we  obtain  the  relation  of  Rodrigues 


362  Polar  Co-ordinates 

This  may  be  derived  also  from  the  equations  of  Schende 


/      \m          /I     i     ,,\  2    //« 

=     (~>  _  (~-^i  -—  rfu  -  nn+ 

2«(n-m)!\l-  p.)    d^1^        > 


~- 

(n-m)!\l-  p. 

which  may  likewise  be  proved  with  the  aid  of  Leibnitz's  theorem.   We 
have  in  fact 

^—[(/x-ir-^+i)^] 

nl  (n-m)\     (n±m]  !  _s 

"  ^  *T(^^  ^-^T^)"!  (m  +  «)  !  (/A        '          (^+    j      ' 
By  differentiating  Conway's  formula  m  times  with  respect  to  t  and 

m 

multiplying  by  (r2  —  £2)2  we  obtain  the  formula 

"  '  0. 


.      -  ,, 

rn+i    »    yry      v     ;    v  '     (n  —  m)\  \rdrj  r 

Making  use  of  the  formula  (C)  we  may  also  write 


Changing  the  sign  of  m  we  have 


This  formula  also  holds  for  m  >  0. 

§  6-28.  Integral  relations.  The  Legendre  functions  satisfy  some  interesting 
integral  relations  which  may  be  found  as  follows : 

Writing  down  the  differential  equations  satisfied  by  Pnm  (/x)  and  Ptk  (/z) 


let  us  first  put  k  =  m  and  multiply  these  equations  respectively  by  Pf 
and  Pnm  and  subtract,  we  then  find  that 


+  (n  -  I)  (n  +  I  +  1)  PnmP^  =  0. 


Integral  Relations  363 

Integrating  between  —  1  and  +  1  the  first  term  vanishes  on  account  of 
the  factor  1  —  /x2  and  so  we  find  that  if  I  ^  n 


Next,  if  we  put  I  =  n  and  multiply  by  Pnfc,  Pnm  respectively  and 
subtract  we  find  in  a  similar  way  that  if  ra2  ^  k2 


To  find  the  values  of  the  integrals  in  the  cases  I  =  n,  k  =  m  we  may 
proceed  as  follows  : 

If  we  multiply  the  first  difference  equation  by  Pmn+1  (/x)  and  integrate 
between  —  1  and  +  1  we  obtain  the  relation 

(n  -  m  +  1)  f1  [P*'B+1  (M)]2  dp  -  (2n  +  1)  P  iiPmn+lPnm  rf/x, 
J-i  J-i 

while  if  we  multiply  it  by  Pmn_!  (/x)  and  integrate  we  obtain  the  relation 

(n  +  m)  f1    [P-n_x  Oz)]2^  =  (2n  -f  1)  f  l   /zP^P'V^. 
J-i  J-i 

Changing  ?i  into  n  —  1  in  the  previous  relation  we  find  that 

(2n  +  1)  (n  -  m)  ['    [Pw«  (^)]2  rf^  =  (2w  -  1)  (n  +  m)  f    [P-n_t  (/x)]2  rf/i. 
J-i  J-i 

But 


Therefore 

[Pnm  (/*)?  ^=  l«-3«  ...  (2m  -  1)*  j^l  -  M«)»  dp  =  ^-j  (2m)  ! 

and  so  f  '   [P.*  (^  ^  =  9-^rT  r  ^  ' 

]-i  r/J     r      2n+l(n~m)\ 

Let  us  next  multiply  the  difference  equations 


f 

(1  -  M2)  —^  =  n^P^  -(n-  m)  Pn- 

by  (1  —  ijP)-lPmn^  and  (1  —  /x2)~1Pww  respectively  and  add. 
Integrating  between  —  1  and  -i-  1  we  obtain  the  relation 


(n  +  m)        [P-W]*  -          -  (n  -  m) 

=  0  if  m  >  0. 


364  Polar  Co-ordinates 

Now 

f1  [Pnm  (^)]\4r~^  I2.32...(2m-  I)2  f1  (I-/*2)" 
=  2. (2m-  1)!. 

MIL.         r  f1     rr»   «,  /     xno       ^M-  1       (?l  -f  m)  ! 

Therefore  [Pnm  (M)]      _     2  =  —  .  V-  —  -    f . 

These  relations  are  of  great  importance  in  the  theory  of  expansions  in 
series  of  Legendre  functions.  See  Appendix,  Note  m. 

§  6« 29.  Properties  of  the  Legendre  coefficients.  If  the  function  /  (x)  is 
integrable  in  the  interval  —  1  <  x  <  1,  which  we  shall  denote  by  the 

symbol  /,  the  quantities 

/>  1 

(3*}  doc  (I) 

are  called  the  Legendre  constants.  If  these  constants  are  known  for  all 
the  above  specified  values  of  n  and  certain  restrictions  are  laid  on  the 
function/(x)  this  function  is  determined  uniquely  by  its  constants.  An 
important  case  in  which  the  function  is  unique  is  that  in  which  the  function 
(I  —  x2)%f(x)  is"  continuous  throughout  /.  To  prove  this  we  shall  show 
that  if  <f)  (x)  =  (I  —  #2)~i</r  (x),  where  iff  (x)  is  continuous  in  /,  then  the 
equations 

f1   <f>(z)Pn(x)dx=0     (n=  0,1,2,...)  (II) 

j-i 

imply  that  $  (x)  =  0. 

The  first  step  is  to  deduce  from  the  relations  (II)  that 

(     <f>(x)xndx=  0     (TI=  0,  1,2,  ...). 

This  step  is  simple  because  xn  can  be  represented  as  a  linear  com- 
bination of  the  polynomials  P0  (x),  Px  (x),  ...  Pn  (x). 

The  theorem  to  be  proved  is  now  very  similar  to  one  first  proved  by 
Lerch*.  The  following  proof  is  due  to  M.  H.  Stonef. 

If  ^  (x)  ^  0  for  a  value  x  =  £  in  /  we  may,  without  loss  of  generality, 
assume  that  $  (£)  >  0,  and  we  may  determine  a  neighbourhood  of  £ 
throughout  which  <f>  (x)  >  m  >  0.  Now  if  A  >  0  the  polynomial 

p  (x)  *==  A  —  %A  (x  —  £)2(x2  -f  1) 

is  not  negative  in  /  and  has  a  single  maximum  at  x  ==  £.  We  choose  the 
constant  A  so  that  in  the  above-mentioned  neighbourhood  of'  f  there  are 

*  Acta  Math.  vol.  xxvn  (1903). 

f  Annals  of  Math.  vol.  xxvii,  p.  315  (1926). 


Theorems  of  Lerch  and  Stone  365 

two  distinct  roots  of  the  equation  p  (x)  =  1  which  we  denote  by  xl ,  x2 ,  the 
latter  root  being  the  greater.  We  thus  have  the  inequalities 
0<  p  (x)  <  1,  -  1  <  x<  xly 

X2<  X  <   1, 

p  (x)  >  1,     <f>  (x)  >  m,        xl  <  x  <  x2y 
ifj  (x)  >  -  M ,  -  Kx<  I, 

w 

where  M  is  a  positive  quantity  such  that  —  M  is  a  lower  bound  for  the 
continuous  function  0  (x). 

Writing  pn  (x)  —  [p  (#)]n,  we  have 

<t>(x)pn(x)dx=0,     n=l,2.  (A) 


L 


On  the  other  hand 


fx,  rx, 

<£  (*0  Pn  (x)  dx>  m\      pn  (x)  dx, 

Jxi  Jxl 

f  '  <f>(x)pn(x)dx>  -  M\  l  (1  -x*)-ldx, 
J-i  J-i 

r1  f1 

<f>  (x)  pn  (x)  dx  >  -  M\      (I-  a;2)-*  dx, 

J  x,  J  xt 

r1  (x*          *          r1 

^  (^)  Pn  (x)  dx>      m  \     pn  (x)  dx  —  M  \     (1  —  #2 

J-l  JXi  J-l 

f*1 
>  m       pn  (a?)  rfa;  —  rrM. 


fx, 

Since        pn  (x)  dx  ->  oo  as  n  ->  oo  we  can  choose  a  number  N  such 

that  the  right-hand  side  is  positive  for  n  >  N.  This  contradicts  (A)  and  so 
we  must  conclude  that  <f>  (x)  =  0  throughout  /. 

Lerch's  theorem  is  that  if  ijj  (x)  is  a  real  continuous  function  and 

/•i 

xn  0  (x)  dx  =  0  for  tt  =  0,  1,  2,  ...  to  oo,  then  iff  (x)  =  0. 
Jo 

By  Weierstrass's  theorem  the  function  i/j  (x)  may  be  approximated 
uniformly  throughout  the  interval  (0,  1)  by  a  polynomial  G  (x).  In  other 
words,  a  polynomial  G  (x)  can  be  chosen  so  that  ifj  (x)  =  G  (x)  -f  86  (x), 
where  |  0  (x)  \  < •  1,  and  8  is  any  small  positive  number  chosen  in  advance. 
Now  if  t/j  (x)  is  not  zero  throughout  the  interval  (0,  1)  we  can  choose  our 

number  8  so  that 

Ji  ri 

o  Jo 

But,  since  G  (x)  is  a  polynomial,  we  have 


366  Polar  Co-ordinates 

Therefore  f  V  (*)  [<A  (*)  ~  8 

Jo 

or  P[<A  (a)]2  da  =  s[  9  (x)  0  (a) 

J  o  Jo 


f1 

Jo 


[ 
o  >o 

This  contradicts  (B)  and  so  we  must  have  0  (x)  =  0.   Putting  x  =  e~* 

roo 

we  deduce  that  if       er*  $  (t)  dt  =  0  f  or  2  >  0,  and  </>  (£)  is  continuous  for 

Jo 
t  >  0,  then  f/>  (J)  =  0. 

EXAMPLES 
1.  When  m  and  n  have  positive  real  paits 


A       Qm  (z)  Qn  (2)  dz  -  *(n  -hi)-  *(m  -h  1)  -2l7r  [(«  -  T)  sin  (\m  -f-  Jw) 


where  ^(z)  =       log  r  (2), 

A   =  (m  -  rt)  (m  -f  n-f  I) 

H  =  /*(}w  +  J,  J'"  +  1)        j  ;,r 

[S.  (*    Dharand  \   (1   Shabde,  Hull   Calcutta  Math   Soc.  v.  24,  177-186  (1932). 


also  that  v\ith  the  same  notation 

Qm  (z)  Qn  (2)  dz  -  t  (m  +  1)  -  *  (w  -f  1) 
((^•ne.sh  Prasad,  Proc   Henarex  Math.  Soc.  v    12,  pp.  33-42,  19.) 


2.   Show  by  means  of  the  relation 


that  when  n  is  a  positive  integer  the  equation  Pn  (ft)  =  0  has  n  distinct  roots  which  all  lie 
in  the  interval  —  1<  /x<  1. 

3.    Prove  that  when  m  and  n  are  positive  integers 

2,m+n-\i  <(m  .   W\u4 

M4-2^m*-nP   ^^P  (2^^  -  .i7^^71;-!. 

i  +  2)       rrn(.)  rw  (zj  a~  -  {mlnl}2  (2m  +  2n  4_  X)  ,. 

[E.  C.  Titchmarsh. 

An  elementary  proof  of  this  formula  is  given  by  R.  0.  Cooke,  Proc.  London  Math.  Soc.  (2), 
vol.  xxin  (1925);  Records  of  Proceedings,  p.  xix. 


Green's  Function  for  a  Sphere  367 

§  6-31.   Potential  function  with  assigned  values  on  a  spherical  surface  S. 
Let  P,  P'  be  two  inverse  points  with  respect  to  a  sphere  of  radius  a. 
If  0  is  the  centre  of  the  sphere  we  have  then 

OP.  OP' -a2, 

and  0,  P,  P'  lie  on  a  line.  The  point  0  is  sometimes  called  the  centre  of 
inversion. 

If  P  lies  inside  the  sphere,  P'  lies  outside ;  if  P  is  outside  the  sphere,  P'  is 
inside.  If  P  is  on  the  sphere,  P'  coin- 
cides with  P.   If  P  describes  a  curve 
or  surface  P'  will  describe  the  inverse 
curve  or  surface  and  it  is  clear  that 
a  curve  or  surface  will  intersect  the 
sphere  at  points  where  it  meets  its 
inverse.  If  a  curve  or  surface  inverts 
into  itself  it  must  intersect  the  sphere 
S  orthogonally  at  the  points  where  it 
meets  it  because  at  these  points  two 
coi  sccutive  inverse  points  lie  on  the 
surface  and  on  a  line  through  0.  This  line  is  then  a  tangent  to  the  surface 
and  a  normal  to  8  at  the  game  point.  If  Ms  is  any  point  on  S  the  triangles 
0PM s,  OMSP'  are  similar,  and  we  have 

OP 


PM8~P'M,' 

If  charges  proportional  to  OP  and  —  OMS  are  placed  at  P  and  P1 
respectively,  the  sum  of  their  potentials  at  any  point  Ms  on  S  will  be  zero. 
Writing  OP  =_r,  PM  =--  R,  P'M  -  R' ',  where  I/  is  any  point,  we  see  that 
the  function 

__  I      a    1 
^PM~P"r*^ 

is  zero  when  M  is  on  S  and  is  infinite  like  -„  at  the  point  P.   We  shall  call 

Zi 

this  function  the  Green's  function  for  the  sphere.  GP}J  is  easily  seen  to  be 
a  symmetric  function  of  the  co-ordinates  of  P  and  M,  £  r  if  JT  is  the 
inverse  of  Jf  we  have 

Oif       OP 
P'M  ^  PM7' 

The  point  P'  is  called  the  electrical  image  of  P  and  (7/>3/  represents  the 
potential  at  M  when  the  sphere  $,  regarded  as  a  conducting  surface  at 
zero  potential,  is  influenced  by  a  unit  charge  at  P.  When  a  becomes 
infinite  and  0  recedes  to  infinity  the  sphere  becomes  a  plane,  P'  is  then  the 
optical  image  of  P  in  this  plane,  and  the  virtual  charge  at  P'  is  equal  and 
opposite  to  that  at  P. 


368  Polar  Co-ordinates 

Now  let  OM  =  r'  and  POM  =  <*>,  then 


2  __  2rr  cos  co, 
__  2r'      cos  co, 


a     a  (ai  +  r2  -  2ar  cos  a>)' 

Let  (r,  0,  <£),  (r,  0',  </>')  be  the  spherical  polar  co-ordinates  of  the  point 
P  and  a  point  Ms  on  the  surface  of  S,  then  the  theorem  of  §  2-32  tells  us 
that  if  a  potential  function  V  is  known  to  have  the  value  F  (#',  <£')  at  a 
point  M8  on  $  then  an  expression  for  V  suitable  for  the  space  outside  S  is 


o  (a2  -f  r2  -  2ar  cos  co) 

while  a  corresponding  expression  suitable  for  the  space  inside  S  is* 

1    fir        r2n        a  (a2  -  r*)F  (#',</>')  sin  6' 
V  (r,  9,  (/>)  -  -i-     rffl'        <Z<A'  -  ~—  7  ~~7  1^         ^T"  - 
^      '  ^;      47rJo       Jo  (a2  +  r2  -  2ar  cos  w)* 

When  the  sphere  becomes  a  plane  the  corresponding  expression  is 

,  ,  ±  /(«',»') 


the  upper  or  lower  sign  being  taken  according  as  z  ^  0.    In  this  case 

/  (x1 ',  y')  =  V  (xf ',  ?/',  0)  is  the  value  of  F  on  the  plane  2=0. 

§  6' 32.  Derivation  of  Poissons  formula  from  Gauss's  mean  value  theorem. 
Poisson's  formula  may  also  be  obtained  by  inversion,  using  the  method  of 
Bocher. 

Let  us  take  P'  as  centre  of  inversion  and  invert  the  sphere  8  into  itself. 

The  radius  of  inversion  is  then  c  =  (r02  —  a2)*  =  -  (a2  —  r2)i,  where  c  is  the 

length  of  the  tangent  from  P'  to  the  sphere,  it  is  real  when  P'  is  outside 
the  sphere  and  imaginary  when  P'  is  within  the  sphere.  (In  Fig.  28 
OP'-r0.) 

Let  Q,  Q'  be  two  corresponding  points  on  $,  then  the  relation  between 
corresponding  elements  of  area  is 

cr    \4 


Writing  dS'  =  a*d£l',  dS  =  a2dfl,  where  d&'  and  dO  are  elementary- 
solid  angles,  we  have 

dtt  =  (    Cp^)  d£l  =  (a2  -  r2)2  (r2  +  a2  -  2ar  cos  w)-2  dQ, 

\U- .  x   v^/ 

where  co  is  the  angle  between  OQ  and  OP. 

*  This  is  generally  called  "Poissoi^s  integral,"  both  formulae  having  been  proved  by  S.D. 
Poisson,  Journ.  £cole  Polyt.  vol.  xix  (1823).  The  formula  for  the  interior  of  the  sphere  had,  how- 
ever, been  given  previously  by  J.  L.  Lagrange,  ibid.  vol.  xv  (1809). 


Poisson's  Formula  and  its  Generalisations  369 

Now  if  V1 'Q>  is  a  potential  function  when  expressed  in  terms  of  the 
co-ordinates  of  Q',  the  function 


P'Q     Q' 

is  a  potential  function  when  expressed  in  terms  of  the  co-ordinates  of  Q, 
consequently  the  mean  value  theorem 


glV6S 


47rF0'  =  a  (a2  -  r2)2  f  F0dQ  [r2  +  a2  -  2ar  cos  «]*, 

CI*  J 


and  since  c.  F0'  =  P'P.  VP,  we  have  crV0'  =  (a2  —  r2)  FP,  and  our  formula 
is  the  same  as  that  derived  from  the  theory  of  the  Green's  function. 

This  method  is  easily  extended  to  the  case  of  hyperspheres  in  a  space 
of  n  dimensions.  The  relation  between  the  contents  of  corresponding 
elements  of  the  hyperspheres  is  now 

*?_'  _  f?'^\n~l  -  (JLVn~2  -  (   cr  Vw~2 
dS-\P'Q)      ~(P'QJ        ~~\a.PQ>        ' 

while  the  relation  between  corresponding  potentials  is 


Writing  the  mean  value  theorem  in  the  form 


the  generalised  formula  of  Poisson  is 


cr 


n 
*«2\n-2  f  /rr\n  f  —  jr 

- )       FP  US'  =  (  -  J      F0  dS  [r2  +  a2  -  2ar  cos  a>]    2 
/  J  VQ-  /  J 


71 

or  VP  J  AS'  =  ±  a"-2  (a2  -  r2)  J  FQ  cZS  [r2  +  a2  -  2ar  cos  cu] "  ^. 

§  6*33.  /Some  applications  of  Gauss's  mean  value  theorem.  The  mean  value 
theorem  may  be  used  to  obtain  some  interesting  properties  of  potential 
functions. 

In  the  first  place,  if  a  function  F  is  harmonic  in  a  region  .R  it  can  have 
neither  a  maximum  nor  a  minimum  in  R. 

If  the  contrary  were  true  and  F  did  have  a  maximum  or  minimum 
value  at  a  point  P  of  R  the  mean  value  of  F  over  a  small  sphere  with 
centre  at  P  would  not  be  equal  to  the  value  at  P.  If  now  the  sphere  is  made 
so  small  that  it  lies  entirely  within  jR,  Gauss's  theorem  may  be  applied 

B  24 


370  Polar  Co-ordinates 

and  we  arrive  at  a  contradiction.  Since  a  function  which  is  continuous 
over  a  region  consisting  of  a  closed  set  of  points  has  finite  upper  and  lower 
bounds  which  are  actually  attained,  we  have  the  theorem  : 

//  a  Junction  is  harmonic  in  a  region  R  with  boundary  B  and  is  con- 
tinuous in  the  domain  R  -|-  B,  the  greatest  and  least  values  of  V  in  the  domain 
R  |-  n  are  attained  on  the  boundary  B. 

One  immediate  consequence  of  the  last  theorem  is  that  if  the  function 
V  is  harmonic  in  R,  continuous  in  R  -f  B  and  constant  on  B  it  is  constant 
on  R  -f  B.  This  theorem  is  important  in  electrostatics  because  it  tells  us 
that  the  potential  is  constant  throughout  the  interior  of  a  closed  hollow 
conductor  if  it  is  known  to  be  constant  on  the  interior  surface  of  the 
conductor.  Another  interesting  consequence  of  the  theorem  is  that  if  the 
function  V  is  harmonic  in  R,  continuous  in  R  \  B  and  positive  on  B  it  is 
positive  in  R  4-  B.  For  if  it  were  zero  or  negative  at  some  point  of  R  the 
least  value  of  V  in  R  would  not  be  attained  on  the  boundary*. 

This  theorem  may  be  restated  as  follows: 

If  \\  and  V2  be  functions  harmonic  in  R  and  continuous  in  R  4-  B, 
and  if  Vv  is  greater  than  (equal  to  or  less  than)  V2  at  every  point  of  B, 
then  \\  is  greater  than  (equal  to  or  less  than)  V2  at  every  point  of  E  -f  B. 

A  converse  of  Gauss's  theorem,  due  to  Koebe,  is  given  in  Kellogg's 
Foundations  of  Potential  Theory,  p.  224. 

§  6'  34.  The  expansion  of  a  potential  function  in  a  series  of  spherical 
harmonics.  If  V  (x,  y,  z)  is  a  potential  function  which  is  continuous 
throughout  the  interior  of  a  sphere  S  and  on  its  boundary,  and  whose 

first  derivatives    ^    ,    ^    ,    ^     arc  likewise  continuous  and  the  second 

ex      en      cz 

derivatives  finite  and  integrable  (for  simplicity  we  shall  suppose  them  to 
be  continuous)  then  V  admits  of  a  representation  by  means  of  Poisson's 
formula  and  it  will  be  shown  that  V  can  be  expanded  in  a  convergent 
power  series  in  the  co-ordinates  x,  y,  z  relative  to  the  centre  of  S.  Writing 

cos  o»  —  cos  6  cos  6'  -f  sin  8  sin  6'  cos  (</>  —  <£')  —  //,, 
we  have 

2         /2\  oo  n    l 

P.  (M)     |  r  |  >  a, 


(a2  4-  r2  -  2 


.,  -     „  - 

(a1  -f  r1  --  2<7rcosoo)*     n 

Substituting  in  the  expressions  for  V  we  may  integrate  term  by  terrr. 
because  the  series  are  absolutely  and  uniformly  convergent  on  account  ot 
the  inequality  |  Pn  (p)  \  <  1. 

*  See  a  paper  by  G.  E.  Ray  nor,  Annals  of  Math.  (2),  vol.  xxm,  p.  183  (1923). 


Expansion  of  a  Potential  Function  371 

We  thus  obtain  the  expansions 


n=0    " 


7-  S  ( 

n=0 

where  in  each  case 


Sn  (6,  ft  =  I    \'  f  'V  (0', 

47Tj()   JO 


sn 


The  function  rnSn  (0,  <f>)  is  called  a,spherical  harmonic  or  solid  harmonic 
of  degree  n,  it  is  a  polynomial  of  the  nth  degree  in  x,  ?/,  z  arid  is  a  solution 
of  Laplace's  equation  because  rnPn  (/*)  is  a  solution. 

The  function  Sn  (9,  <f>)  is  called  a  surface  harmonic,  it  may  be  expressed 
in  terms  of  elementary  products  of  type  Pnm  (cos  0)  e*tn*  by  expanding 
Pn  (fji)  in  a  Fourier  series  of  type 

Pn(/A)  =    2    Fnm(0,0')e"»  <+-*'>. 

in—  —n 

By  expanding  rnPn  (p)  in  a  series  of  this  form  and  substituting  in 
Laplace's  equation  (in  polar  co-ordinates)  we  get  a  series  of  typeSCfwe*m(*""*/) 
each  term  of  which  must  be  separately  zero,  consequently  each  term  in  our 
expansion  of  rnPn  (p,)  is  a  solution  of  Laplace's  equation  and  is  a  poly- 
nomial of  degree  n  in  x,  y  and  z.  Similarly,  if  r',  0'  ,  c/>'  are  regarded  as 
polar  co-ordinates  of  a  point  (x'9  yr,  z'),  r'HPn  (IJL)  is  a  solution  of  Laplace's 
equation  relative  to  the  co-ordinates  of  this  point.  We  infer  then  that 

Fn™  (0,  8')  -  AnmPnm  (cos  0)  Pn-m  (cos  0'), 
where  Anm  is  a  constant  to  be  determined. 
We  thus  have  the  result  that 


Sn(0,<j>)=    S    £nmP«m  (cos  6)  e"»*, 

m=*—n 

where    Bn™  -   .—  ^nm  f  f  ^ F  (0f,  <f>')  Pn~m\cos  9')  er™*'  sin  0'd9'dcf>'. 

477  J  0  J  0  • 

To  determine  the  constant  Anm  we  consider  the  particular  case  when 

V  =  rnPMm  (cos  9)  eim*, 
F  (9',  </>')  =  anPnm  (cos  0')  e*™+\ 
then  (2i/+l)  S,  (fl,  <£)  -  anPw^  (cos  (9)  e'™*    i/  -  n 

=  0  ^=^w, 

and  consequently 

,2™  4-  1 

X-  -4^ 

or      ^4nw  —  (—  )m. 

24-2 


Anm  ^  f  ""  Pnm  (cos  0')  Pn-«  (cos  9')  sin  0'd9'd<f>' 
Jo  Jo 


372  Polar  Co-ordinates 

Hence  we  have  the  expansion 

Pn  (p)  -    S    (-)mPnm  (cos  6)  Pn~™  (cos  0')  e"»<*-*'>. 

m**—n 

§  6*35.  Legendre's  expansion.  Transforming  the  last  equation  with  the 
aid  of  the  relation  of  Rodrigues, 

U  ~~  m 


-m  /..\  __ 
- 


we  obtain  Legendre's  expansion* 

00  (rvi      _  .     Yf)  \    t 

Pn  (p)  =  2   S      —  -  -  j  Pnm  (cos  6)  Pn»  (cos  0')  cos  m  (<£  -  f  ) 

m—l  \'^    '     '"7  i 

+  Pn  (cos  0)  Pn  (cos  0'), 
and  the  expression  for  Bnm  may  be  written  in  the  alternative  form 

1    (rt  — 


f  f  w 

1  10 


One  simple  deduction  from  the  expansion  for  Sn  (6,  <f>)  is  that  a  simple 
expression  can  be  obtained  for  the  mean  value  of  Sn  (9,  </>)  round  a  circle 
on  the  sphere.  Let  the  circle  in  fact  be  0  =  a,  then  the  mean  value  in 
question  is  obtained  by  integrating  our  series  for  Sn  (9,  </>)  between  0=0 
and  $  =  2-n-  and  afterwards  dividing  by  2ir.  The  result  is  that 


tin  (0,  </>)  =  Bn»Pn  (cos  a). 

Now  when  0=0,  Pnm  (cos  9)  =  0  except  when  m  =  0,  and  then  the 
value  is  unity,  hence 

Sn  (0,  0)  =  ,Bno, 


and  so  Sn  (6,  <f>)  =  flfn  (0,  0)  Pn  (cos  «), 

where  the  coefficient  Sn  (0,  </>)  is  the  value  of  Sn  (9,  <f>)  at  the  pole  of  the 
circle.  This  theorem  may  be  extended  so  as  to  give  the  mean  value  of  a 
function  /  (9,  </>),  which  can  be  expanded  in  a  series  of  type 


The  result  is  ni=0 


n=0 

If  the  analytical  form  of  the  function /is  not  given,  but  various  graphs 
are  available,  the  present  result  may  sometimes  be  used  to  find  the 
coefficients  in  the  expansion 

/(0,<£)=  2  C'nPn(cos0)+  S    E   Pwm(cosfl)[-4nmcosm^  +  5nmsinm^]. 

n=0  n==l  m=0 

To  use  the  method  in  practice  it  is  convenient  to  have  a  series  of  curves 
in  which  /  is  plotted  against  <f>  for  different  values  of  0  and  a  series  of 
curves  in  which  /  is  plotted  against  9  for  different  values  of  <f>.  The  two 
*  Legendre,  Hist.  Acad.  Sci.  Paris,  t.  n,  p.  432  (1789). 


Expansion  of  a  Polynomial  373 

meridians  <f>  =  /3  and  <f>  =  /?  4-  TT  may  be  regarded  as  one  great  circle  with 


Mean  values  round  "parallels  of  latitude"  for  which  6  has  various 
constant  values  will  give  linear  equations  involving  only  the  coefficients  Cn . 

Since  Pn  (0)  =  0  when  n  is  odd  and  Pnm  (0)  =  0  when  n  is  even  and 
m  is  odd,  the  mean  values  round  meridian  circles  will  give  equations 
involving  only  the  coefficients  Anm  and  Bnm  in  which  both  m  and  n  are 
even,  but  terms  of  type  Cn  will  also  occur.  To  illustrate  the  method  we 
shall  suppose  that  the  function  /  (9,  <f>)  is  of  such  a  nature  that  spherical 
harmonics  of  odd  order  or  degree  do  not  occur  in  the  expansion  and  that 
a  good  approximation  to  the  function  may  be  obtained  by  taking  terms 
of  orders  and  degrees  up  to  n  =  4  and  m  —  4.  We  have  then  to  determine 
the  nine  coefficients  <70,  C2,  64,  A22,  £22,  ^444,  J544,  ^442,  J342.  Three  of  these 
may  be  determined  from  the  mean  values  of  /  round  parallels  of  latitude, 

say  0  =  ^,  0  =  =,  0  =  -.    Two  equations  connecting  A22,  A^,  A42  may  be 
2  o  u 

obtained  from  the  mean  values  of  /  round  the  meridians  </>  =  0,  TT  and 

<f>  =  ~,  -0- ,  while  two  equations  involving  jB22,  JS42,  A£  may  be  obtained 

£      2i  . 

from  the  mean  values  round  the  circles  <A  =  ~  ,    -.-  :  J>  =  —r ,  c6  =  — - . 

^       4       4      r        4      T         4 

Further  equations  may  be  obtained  from  the  mean  values  round  the  circles 
,,        77     ITT     ,       77     4?r     ,        2?r     STT     ,        5?r     HTT 
^  =  6'  T;<^  =  3'  "3;*="3"'  '3"^^   6  J  ~6~- 

Having  found  CQ,  (72,  C4  from  the  first  three  equations  and  having 
expressed  A22,  A^,  JS22,  542  in  terms  of  Af  with  the  aid  of  the  next  four, 
two  of  the  last  set  of  equations  can  be  transformed  into  equations  for 
Af  and  Bf. 

When  the  two  sets  of  curves  have  been  drawn  the  mean  values  of  / 
round  the  different  circles  may  be  found  with  the  aid  of  a  planimeter. 

§  6*36.  Expansion  of  a  polynomial  in  a  series  of  surface  harmonics, 
When  rnF  (9,  <f>)  is  a  polynomial  of  the  nth  degree  in  x,  y,  z  the  expansion 
of  F  (0,  $)  in  a  series  of  surface  harmonics  may  be  obtained  in  an  elementary 
wav  by  using  the  operator  V2.  Let  us  write  rnF  (9,  <f>)  =  fn  (x,  y,  z).  The 
first  step  is  to  determine  a  polynomial  /n_2  (x,  y,  z)  such  that 

fn  (X>  y,  Z)  -  ?*2/n_.2  (X,  y,  Z) 

is  a  solution  of  Laplace's  equation  of  type  rnSn  (9,  <f>).  The  equation 

V2[/n-r*/n_2]=0 

gives  just  enough  equations  to  determine  the  coefficients  in/n_2.  To  show 
that  the  determinant  of  this  system  of  linear  equations  does  not  vanish 
we  must  show  that  y2  rrzf  ]  =t  o 


374  Polar  Co-ordinates 

If,  however,  r2/n_2  were  a  spherical  harmonic  of  degree  n  we  should  have 
(^2/n_2)/w-2^  —  0  when  integrated  over  the  spherical  surface,  because /n_2 

can  be  expressed  in  terms  of  surface  harmonics  8m  (9,  </>)  of  degree  less 
than  n.    But  this  equation  is  impossible  unless /n_2  vanishes  identically. 

Having  found  /n_2  we  repeat  the  process  with  /n_2  in  place  of  fn  and 
so  on.  We  thus  obtain  a  series  of  equations 

f     __   r2f          __   r2.Qf 

J  n          '  J  n—2          '    ^n  J 

f          —  T2f          =  T2S 

from  which  we  find  that 

When  n  =  2m,  where  w  is  an  integer  and/n  =  /,  the  spherical  harmonics 
are  determined  by  the  system  f)f  equations* 

V2"-/-  (2m,  2)  (2m  4-  1,3)S0, 

V2w-2/  =  (2m,  4)  (2m  4-  1,  5)  r2#0  -I-  (2m  -  2,  2)  (2m  4-  3,  7)  r2S2, 
V2m-4y  =  (2m,  6)  (2m  4-1,7)  r4/S0  4-  (2m  -  2,  4)  (2m  -f  3,  9)  r*82 

4-  (2m  -  4,  2)  (2m  4   5,  11)  r4/SY4, 

where  (a,  6)  =  a  (a  —  2)  (a  —  4)  ...  6. 

Solving  these  linear  equations  we  find  thatf 
r'^Sw  (2m  -  2k,  2)  (2m  4-  2k  4-  1,  4fc  4-  3) 

r4 

+  2.4~(4]fc  -" 

where  02fc  (r,  y,  2)  =  V2m-2kf  (x,  y,  z). 

The  equivalence  of  the  two  expressions  for  S  is  a  consequence  of 
Hobson's  theorem  (§  6-22). 

There  is  a  corresponding  theorem  for  a  space  of  n  dimensions.  The 
fundamental  formula  for  the  effect  of  the  operator 

is  Vw2  (r2%)  -  2p  (2p  -f-  2q  4-  n  -  2)  r2*>~%  -f  r*»Vn*vQ, 

where  r2  =  o:^  4-  x22  4-  ...  xn2,     t;v  =  VQ  (xl9  x2,  ...  xn). 

*  If  VQ  (x,  y,  z)  is  a  rational  integral  homogeneous  function  of  degree  m,  we  have 

V2  (r*%)  =  2p  (2p  +  2q  +  1)  r2?-2^  -f  r2P  V2v0. 

Hence  if  V2vQ  =  0  the  effect  of  successive  operations  with  V2  is  easily  determined. 
f  G.  Prasad,  Math.  Ann.  vol.  LXXII,  p.  435  (1912). 


Legendre  Functions  of  i  cot  6  375 

The  equations  are  now 
Vn**/=  (2m,  2)  (2m  +  n  -  2,  n)  S0, 
Vn2™-2/  -  (2m,  4)  (2m  -f  7i  -  2, 7i  -f  2)  r2S0  4-  (2m  -  2,  2)  (2m  +  n,  n  -f  4)  r2S2 , 

r2*^  (2m  -  2i,  2)  (2m  -f-  2fc  +  n  -  2,  4fc  +  n) 

r2\7  2  r4V  4  1 

*  4_  _  _  n  —  V2m-2fc/' 

2(4fc  +  n-  4)  '  2.4(4fc  +  ra-  4)  (4i  +  rc-  6)      '"J  ' 

__  r4*+*-2    _      /  a     a         a\  2_n 

-  (ifc  -  4  +  w,  n  -  2)  fe  Va^'  8^2 '  '"  dxj  T      ' 
where  02fc  (^ ,  o:2 ,  . . .  xn)  -  Vn2w|-2*/  (^ ,  a?2 ,  . . .  a:n), 

and  /  is  a  homogeneous  polynomial  of  degree  2m. 

§  6-41.  Legendre  functions  and  associated  functions.  It  should  be 
observed  that  Laplace's  equation  possesses  solutions  of  type 

rnPnm  (/A)  e*tm*,     r»Qnm  (p.)  e±tm*, 

when  n  and  m  are  any  numbers.  It  is  useful,  therefore,  to  have  definitions 
of  the  functions  Pnm  (/x)  and  Qnm  (/z)  which  will  be  applicable  in  such  cases 
and  also  when  fi  is  not  restricted  to  the  real  interval  —  1  <  /*  <  1 . 

The  need  for  such  definitions  will  appear  later,  but  one  reason  why 
they  are  needed  may  be  mentioned  here. 

In  an  attempt  to  generalise  the  method  of  inversion  for  transforming 
solutions  of  Laplace's  equation*  it  was  found  that  if 

Y  -  Jl^-?2         7  -  -  t    r*  +  a~        Z  =  -  °*-  (I) 

2(x-M/)'  2(x-iy)'  x-iy'    (> 

an(J  if  /  (X,  Y,  Z)  is  a  solution  of  Laplace's  equation  in  the  co-ordinates 
X,  Y,  Z,  then  (x-iy)-lf(X,  Y,Z) 

is  a  solution  of  Laplace's  equation  in  the  variables  x,y,z.  Introducing 
polar  co-ordinates,  we  find  that 

R  =  iae1*,     r  =  iae1*,     sin  0  =  cosec  9. 

The  standard  simple  solutions  of  Laplace's  equation  give  rise,  then,  to 
new  simple  solutions  of  type 

(sin  0)~*  Pnm  (i  cot  9)  ct(n+i>*  r~l+m, 

and  we  are  led  to  infer  the  existence  of  reciprocal  relations  between 
associated  Legendre  functions  with  real  and  imaginary  arguments  and  of 
more  general  relations  when  the  arguments  are  complex  quantities  or  real 
quantities  not  restricted  to  the  interval  —  1  <  z  <  1 . 

Definitions  of  the  associated  Legendre  functions  Qnm  (z)  for  all  values 

*  Proc.  London  Math.  Soc.  (2),  vol.  vii,  p.  70  (1908). 


376  Polar  Co-ordinates 

of  n,  m  and  z  have  been  given  by  E.  W.  Hobson*  and  by  E.  W.  Barnesf. 
The  definitions  adopted  by  Barnes  are  as  follows  : 

Let  z  =  x  -h  iy,    w  =  log  -  —  , 

25  —   1 

»  2»r(l-ra-.s)  jmu, 

e 


then,  if  |  arg  (z  —  1)  |  <  TT, 

^n™  (^)  ==  —  sin  7&7T    y  (m,  n,  5)  (2  —  I)8  d#, 

where  the  integral  is  taken  along  a  path  parallel  to  the  imaginary  axis  with 
loops  if  necessary  to  ensure  that  positive  sequences  of  poles  of  the  integrand 
lie  to  the  right  of  the  contour,  and  negative  sequences  to  the  left.  Also 

Qnm  (z)  =      e~m 


where  Im  =  TT  cosec  n7r.Pnm  (z),  and  the  upper  or  lower  sign  is  taken  in  the 
exponential  factor  multiplying  Im  according  as  y  ^  0. 

The  functions  Pnm  (z),  Qnm  (z)  are  not  generally  one-valued.  To  render 
their  values  unique  a  barrier  is  introduced  from  —  oo  to  1.  When'w  is 
not  a  positive  integer  and  z  is  not  on  the  cross-cut,  Pnm  (z)  is  expressible 
in  the  form 

Pnm  (^  -  p  -~_  —  -}  AmwF{  -  n,  n  +  1;  1  -  m;  J  (1  -  «)}, 

where  F  (a,  b  ;  c  ;  x)  denotes  the  hypergeometric  function  or  its  analytical 
continuation.  This  formula,  which  gives  a  convergent  series  when 
|  1  —  z  |  <  2,  shows  that  z  =  1  is  a  singular  point  in  the  neighbourhood  of 
which  Pnm  (z)  has  the  form 

(,_  1)-**  {(70  +  ^(2-  1)+...}. 
Under  like  conditions 

—  2Qnm  (z)  .  sin  rnr.  F  (—  m  —  ri) 


*   -f    1 

where,  as  before,  ew  =  -  -  . 

z  —  1 

The  definition  of  Qnm  (z)  given  by  Barnes  differs  from  that  given  by 
Hobson,  the  relation  between  the  two  definitions  being  given  by  the 
formula 

sin  HTT  [Qnm  (z)]B  =  e~im"  sin  (n  +  m)  7r.[Qnm  (z)]H. 

*  Phil.  Trans.  A,  vol.  CLXXXVII,  p.  443  (1896). 
4-  n».**f    IM.~*»    ™%i   WVTV    ^  OT  /icmo\ 


Relations  between  the  Functions  377 

It  follows  from  the  definitions  that 

PB™  (-  z)  =  e*»-*  P.-  (z)  -  2  5£^T  £„«  (2), 

#nm(-2)  =   -<?„">  (Z).e±"", 
P"-,-!  (Z)  =  Pnm  (2), 
£"•_„_!  (Z)  =  <?„»»  (2)  -  7T  COt  Tfor.P,"  (Z), 

Vm  <*)  ."1  (  sin 


g»-m  (z)  r  (m  -  n)  =  &•»  (z)  r  (-  m  -  »). 

When  m  =  0,  or  when  m  is  an  integer,  Pnm  (z)  has  no  singularity  at 
z=l.  This  is  evident  from  the  expression  for  Pnm  (z)  in  terms  of  the 
hypergeometric  function  in  the  cases  when  m  is  negative  or  zero  and  may 
be  derived  from  the  formula 

-  rl~  m  +  n) 


n          -n  (i+       -_ 

in  the  case  when  m  is  positive.   We  add  some  theorems  without  proofs. 

1°.   The  nature  of  the  singularity  of  Pnm  (z)  at  z  =  —  1  may  be  in- 
ferred from  the  formula 


+  e~m  r  (-  nmr+  n)  F{1  ~  m  +  n'  ~  m  ~  n>  l  ~ 

where  0  =  |  (1  -f  z).   When  m  is  a  positive  integer, 


x  F{m  —  n,  m  +  n+  l;m+  1;  J  (1  —  «)}  .......  (A) 

2°.   When  in  addition  n  is  an  integer  there  are  three  cases: 

(1)  0  <  n<  m.  In  this  case  Pnm  (z)  =  0  but  r  (1  +  n  —  m)  Pnw  (2)  is  a 
solution  of  the  differential  equation. 

(2)  n>  m.   In  this  case  the  formula  (A)  is  valid. 

(3)  n  <  0.   In  this  case,  if  —  n  >  m, 

p  m  (z)  _  fc2  _   Dim  T  (m  -  71)  __ 

^n    W-iz        i)      2T(-m-n)r(m+l) 

x  ^{w  —  n,  ra-fw+  1  ;  m  +  1;  £(1  —  z)}. 

3°.    If  -  n  <  m,  Pnm  (2)  =  0,  but  F  (-  m  -  n)  Pnm  (z)  is  a  solution  of 
the  differential  equation. 


378  Polar  Co-ordinates 

4°.  When  m  =  0  and  n  is  not  an  integer  and  is  not  zero, 


=  s  rji  ~  nll  (?L±J _+J)  0< 

x  {log  0  -  20  (1  4-  0  4-  ^  (*  -  ft)  +  ^  (ft  +  1  -f  OK 
where  0  =  i  (1  4-  3)   and   0  (?/)  =  --.-  log  F  (%)• 

2    V  /  V     \      /  fa        fe          \      / 

Hence  Pn(z)  has  a  logarithmic  singularity  at  x  —  —  1,  at  which  it 
becomes  infinite  like 

TT~I  sin  UTT .  F  {  —  n,  n  -f  1 ;  1 ;  0}  log  0  -f  a  power  series  in  0. 

5°.  When  m  —  0,  we  have  seen  that  Pn  (z)  has  a  cross-cut  from  —  oo 
to  —  1 ;  when,  however,  \m  is  not  an  integer  and  not  zero,  Pnm  (z)  has  a 
cross-cut  from  —  oo  to  1,  and  is  therefore  not  defined  by  the  preceding 
formulae  when  —  1  <  3  <  1.  It  is  convenient  to  have  a  single  value  of  the 
function  in  this  interval,  and  one  which  is  real  when  m  and  n  are  real. 
It  is  therefore  assumed  that  as  e  ->  0  and  —  1  <  x  <  1, 

Pnm  (x)  -~  lim  e*mnl  Pnm  (x  -f  €i)  -  lim  e-^mtpnm  (x  _  €i) 
1          /I  -h  x\ 


-  7/         ~~  ,-  m;     - 


n™  (x)  -  I  lim  {Qn>*  (x  +  ei)  +  Qn™  (x  -  €i)} 

=    _         _  -  ^   1^..   .         [0Cm>  (X)  + 

2  sin  H,TT  I  (—  m  -  n)1  ' 


where 


6°.  The  function  Qnm  (x)  has  a  cross-cut  between  —  1  and  1.  For  values 
of  z  for  which  |  z  \  >  1  the  function  can  be  expanded  in  a  convergent  power 
series  in  1/3.  If  |  arg  (z  ±  1)  |  <  TT  and  (2-  -  l)irn  -  (z  -  l)*m  (2  +  l)im, 

/    ,n  /x      sin>  +  w)  ^  r  («•  +  m  4  1)  T  (J)  (2a  -  l)Jw 


x  F(Jn-f  Jm  -f  1,  jn-f  |m  +  J;w  +  f  ;«-2). 

The  values  for  cases  in  which  |  3  |  <  1  may  be  deduced  by  analytical 
continuation  of  the  hypergeometric  function  and  use  of  the  foregoing 
definition  when  z  is  real. 


Reciprocal  Relations  379 

§  6*42.   Reciprocal  relations*.   Barnes  has  shown  that  the  power  series 
in  l/z  can,  under  the  foregoing  conditions,  be  expressed  in  the  form 


O  ™(z\  ~  C 

Vn      \Z)  ~  O   -- 


;  T-—   , 

i  —  z  j 


sin  (n  -f  m)  TT  T  (n  +  m  +  1)  T  (4) 
°  ""    '"sin  nit  "        2»"  r  (n  -f  |)       ' 

Putting  z  =  i  cot  (9,  we  have 
(>nm  (i  cot  0)  -  Ci-n-1  sin**1  0 

x  ^"{|  (n  +  m  +  1),  \  (n  —  7>i  +  l);n  -f  |;  sin20}. 
Now 
F  {J  (n  -f  m  4-  1),  I  (n  -  m  4-  1)  ;  n  4-  f  ;  sin2  9} 

=  F  {n  -f  m  +  1,  n  -  m  +  1  ;  n  +  jj  ;  sin2  £0} 
-  (cos  ifl)-2"-1  .F  [m  +J,J-w;n-f|;  sin2  10]. 
Therefore 

Qn«  (i  cot  0)  -  CT11-1  2"  f*  (sin  0)i  (cot  |0)~W"J 

x  JP  [m  +  A,  i  -  m  ;  n  +  |  ;  sin  J0]. 
But 


3-  (cot  |0)  n  4  F  {m  -f  £,  J  —  M;  n  -f-  f ;  sin2  £0}. 
Therefore 

Qnm  (i  cot  0)  = . ^7 — — (1  sin  0)*  P"n-*  (cos  0). 

sin  n-n .  1  (—  m  —  n)   *  -m-t 

Writing  —  m  —  |  in  place  of  ft  and  —  n  —  \ ,  in  place  of  m,  the  formula 
becomes 

Q~"~\  (i  COt  0) -^4~ — Tx  (1  sin  0)*  PnW  (COS  0). 

^_m_i  v  /      cos  m7T  Y  (m  -f-  n  -f  1) 

Again,  Barnes  has  shown  that  when  |  1  —  z2  |  >  1, 


-f  C2  (2*  -  l)n^    i  (m  -  7i),  £  (-  m  -  n);     -  n; 

I  1 

where 

2—(--)  2Tj(n+  |)       . 


*  Judging  from  a  conversation  with  Dr  Barnes  in  1908  he  had  at  that  time  noted  at  least  one 
explicit  reciprocal  relation  between  the  functions  Pnm(z)  and  Qnm(z). 


380  Polar  Co-ordinates 

consequently,  using  again  the  transformations  of  the  hypergeometric 
series,  we  find  that 


P™  (i  cot  6)  =  (2*  cosec  0)~ 

x  .F{£-hra,  £  —  m;  n  -f-  f  ;  sin2  £0} 

+  rlrT^1!)  6l""(cot  ^)n+*  -**  tt~  ^J  +  ^;i-^;sin2^} 

=  -  sin  (w  +  I)  TT.  r  (w  +  w  +  1)  (27r  cosec  0)~*  lim  Ql*l\  (cos  0  -  fe), 

7T  e->0 

and  so 


P"n"*  (i  cot  0)  =  -  -  sin  UTT  .  T  (-  m  -  n)  (2n  cosec  0)-*  lim  Qn™  (cos  0  -  t€) 

~~w"~*  77  e^Q 

=  -  «,-  ------  r  —  v--—'/  ---  -—  x-  (27r  cosec  0)~*  lim  ^nm  (cos  0  -  fc). 

r(l  4-  m  -f  n)  sm(m  +  n)ir  *->o 

This  is  very  similar  to  the  reciprocal  formula  obtained  by  F.  J.  W. 
Whipple*,  which  may  be  written  in  the  form 

n)        ^ 


nm  (cosh  a)  -  »___m      n         ^ 
"    v  ;  *         -m~* 


EXAMPLES 

1.  Prove  that  when  n  is  a  positive  integer 

{(/t2  -  1)n  log  *  (1  +  ")}  -  Pn 

[A.  E.  JoUiffe,  Mess,  of  Math.  vol.  XLIX,  p.  125  (1919).] 

2.  Prove  that  if  2x  =  <i  +  t~i, 


i  i,  1,  1  -  I) 

=     t*    (2  log  2  -  i  log  t)  f  (|,  i;  1;  0 


I)' 


Prove  also  that 


-      1  [(4  log  2  -  4  -  log  0  ^  (?,  i;  1;  0  -  4  {(f)2  (J  -  J)  * 

+  i  [(f  )2  ~  (S)2]  (J  +  *  -  i  -  i)  t*  +  ...}]. 
[H.  V.  Lowry,  Phil.  Mag.  (7),  vol.  n,  p.  1184  (1926).] 

*  Proc.  London  Math.  Soc.  (2),  vol.  xvi,  p.  301  (1917). 


Potentials  of  Degree  n  +  \  381 

3.  If  K  is  the  quarter  period  of  elliptic  functions  with  modulus  k  and  complementary 
modulus  k*  =  (1  -  &a)i,  prove  that 


V         " 

K  =  ^ 


W 


,P- 


2p*"ni-jfc« 


(1*-*)  P  (**  i;  *  ;  (T+l?) 


The  last  series  is  recommended  by  Lowry  for  the  calculation  of  jfiT  when  k  is  nearly  1. 

4.  Prove  that  if  n  is  a  positive  integer  the  equation  Pn-f  (z)  =  0  has  no  root  which  lies 
in  the  range  1  <  z  <  3. 

5.  Prove  that  if  2n  +  1  4=  0 


6.   Prove  that  if  n  is  a  positive  integer 

Pn(